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MECHANICAL piNCIPLE 

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ARCHITECTURE. 



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HENRY MOSELET, M.A. F.R.S. 

CHAPLAIN IN ORDINARY TO THE QUEEN, CANON OF BRISTOL, VICAR OP OLVESTON ! 

CORRESPONDING MEMBER OF THE INSTITUTE OF FRANCE, AND FORMERLY PROFESSOB 
OF NATURAL PHILOSOPHY AND ASTRONOMY IN KING'S COLLEGE, LONDON. 



Second American from Second London Edition 



WITH ADDITIONS BY 

D. H . M A H A N , L 

U. S. MILITARY ACADEMY 

WITH ILLUSTRATIO 




NEW YORK: 

JOHN WILEY k SON, PUBLISHERS, 

2 Clinton Hall, Astor Place. 

1869. 



ExrrarD according \o Act of Congress, !n the year 18M, \j 

WILEY * H ALSTED, 

Is the C.r rk's Office of the District Court of the United States, for the Soatha l 

of New York. 



(N ■XCHAWOi 

Columbia. Umv 
MAE 20 \m 



-2fl$ 




EDITOR'S PREFACE. 



The high place that Professor Moseley occupies in the 
scientific world, as an original investigator, and the clear- 
ness and elegance of the methods he has employed in this 
work have made it a standard text book on. the subjects it 
treats of. In undertaking its revision for the press, at the 
request of the publishers of this edition, it has been deemed 
advisable, in view of the class of students into whose hands 
it may fall, to make some slight addition to the original. 
This has been done in the way of Notes thrown into an 
Appendix, the matter of which has been gathered from 
various authorities ; but chiefly from notes taken by the 
editor, whilst a pupil at the French military school at Metz, 
of lectures delivered by General Poncelet, at that time, 1829, 
professor in that school. It is a source of great pleasure to 
the editor to have this opportunity of publicly acknowledg- 
ing his obligations to the teachings of this eminent savan, 
who is distinguished not more for his high scientific attain- 
ment, and the advancement he has given to mechanical 
science, than for having brought these to minister to the 
wants of the industrial classes, the intelligent success of 
whose operations depends so much upon mechanical science, 
by presenting it in a form to render it attainable by the most 
ordinary capacities. 

Hi 



it editor's PREFA<" 

Tlie editor would remark that he has carefully refrained 
from making any alterations in the text revised, except cor- 
- aphical :id in one instance where, 

from a repetition of apparently one of these, he apprehended 
gome difficulty might be offered to the student if allowed 
to remain exactly as printed in the original* 

M n.iTiRT Academy, 
West Point March 8, 1856. 



PREFACE TO THE SECOND EDITION. 



I have added in this Edition articles : — first, " On the 
Dynamical Stability of Floating Bodies ;" secondly, " On 
the Rolling of a Cylinder ;" thirdly, " On the descent of a 
body npon an inclined plane, when subjected to variations of 
temperature, which would otherwise rest npon it ;" fourthly, 
" On the state bordering upon motion of a body moveable 
about a cylindrical axis of finite dimensions, when acted 
upon by any number of pressures." 

The conditions of the dynamical stability of floating 
bodies include those of the rolling and pitching motion of 
ships. The discussion of the rolling motion of a cylinder 
includes that of the rocking motion to which a locomotive 
engine is subject, when its driving wheels are falsely 
balanced, and that of the slip of the wheel due to the same 
cause. The descent of a body upon an inclined plane 
when subjected to variations in temperature, which other- 
wise would rest upon it, appears to explain satisfactorily the 
descent of glaciers. 

The numerous corrections made in the text, I owe chiefly 
to my old pupils at King's College, to whom the lectures 
of which it contains the substance, were addressed. For 



vi PREFACE TO FBH SECOND EDITION. 

BeveraJ important ones I am, however, indebted to Mr. 
Robinson, Blaster of the School for Shipwrights' Apprentices, 
in Ber Majesty's Dockyard, Portsea ; to whom I have also tc 
express my warm acknowledgments for the care with 
which he has corrected the proof sheets whilst going through 
the press. 

May, 1865 



PREFACE. 



In the following work, I have proposed to myself to apply 
the principles of mechanics to the discussion of the most 
important and obvious of those questions which present 
themselves in the practice of the engineer and the architect ; 
and I have sought to include in that discussion all the 
circumstances on which the practical solution of such ques- 
tions may be assumed to depend. It includes the substance 
of a course of lectures delivered to the students of King's 
College in the department of engineering and architecture, 
during the years 1840, 1841, 1842 * 

In the first part I have treated of those portions of the 
science of Statics, which have their application in the theory 
of machines and the theory of construction. 

In the second, of the science of Dynamics, and, under this 
head, particularly of that union of a continued pressure with 
a continued motion which has received from English writers 
the various names of " dynamical effect," " efficiency," " work 
done," " labouring force," " work," &c. ; and " moment 
d'activiteV' "quantite d'action," "puissance mecanique," 
" travail," from French writers. 

Among the latter this variety of terms has at length given 
place to the most intelligible and the simplest of them, 

* The first 1*70 pages of the work were printed for the use of my pupils in the 
year 1840. Copies of them were about the same time in the possession of 
several of my friends in the Universities. 



Vlll I'Kl.lAi I.. 

M travail." The English word "work" is the obvious trans- 
lation of "travail," and the use of it appears to be recom- 
mended by the Bame considerations. The work of overcoming 
a pressure of one pound through a space of one foot has, in 
this country, been taken as the unit, in terms of which any 
other amount of work is estimated; and in France, the work 
of overcoming a pressure of one kilogramme through a space 
of one metre. M. Dupin has proposed the application of the 
term dyname to this unit. 

I have gladly sheltered myself from the charge of having 
contributed to increase the vocabulary of scientific words, 
l.v assuming the obvious term "unit of work" to represent 
concisely and conveniently enough the idea which is attached 
to it. 

The work of any pressure operating through any space is 
evidently measured in terms of such units, Dy multiplying 
the number of pounds in the pressure by the number of feet 
in the space, if the direction of the pressure be continually 
that in which the space is described. If not, it follows, by 
a simple geometrical deduction, that it is measured by the 
product of the number of pounds in the pressure, by the 
number of feet in the projection of the space described,* 
upon the direction of the pressure; that is, by the product 
of the pressure by its virtual velocity. Thus, then, we 
conclude at once, by the principle of virtual velocities, that 
it' a machine work under a constant equilibrium of the 
pressures applied to it, or if it work uniformly, then is the 
i gate work of those pressures which tend to accelerate 
Its motion equal to the aggregate work of those which tend 
to rrard it; and, by the principle of vis viva, that if the 
machine do not work under an equilibrium of the forces 
impressed apon it, then is the aggregate work of those which 
tend to accelerate the motion of the machine greater or less 

* If the direction of the pressure remain always parallel to itself, the space 
described may he any finite space ; if it do not, the space is understood to be 
so small, that the direction of the pressure may be supposed to remain parallel 
to iteelf whilst that 6pace is described. 



PREFACE. IS 

than the aggregate work of those which tend to retard its 
motion by one half the aggregate of the vires vivce acquired 
or lost by the moving parts of the system, whilst the work is 
being done upon it. In no respect have the labours of the 
illustrious president of the Academy of Sciences more con- 
tributed to the development of the theory of machines than 
in the application which he has so successfully made to it of 
this principle of vis viva* In the elementary discussion of 
this principle, which is given by M. Poncelet, in the intro- 
duction to his Mecanique Tndustrielle, he has revived the 
term vis inertice (vis inertice, vis insita, Newton), and, 
associating with it the definitive idea of a force of resistance 
opposed to the acceleration or the retardation of a body's 
motion, he has shown (Arts. 66. and 122.) the work expended 
in overcoming this resistance through any space, to be 
measured by one half the vis viva accumulated through the 
space ; so that throwing into the consideration of the forces 
under which a machine works, the vires inertia of its moving 
elements, and observing that one half of their aggregate vis 
viva is equal to the aggregate work of their vires inertice, it 
follows, by the principle of virtual velocities, that the differ- 
ence between the aggregate work of those forces impressed 
upon a machine, which tend to accelerate its motion, and 
the aggregate work of those which tend to retard the motion, 
is equal to the aggregate work of the vires inertice of the 
moving parts of the machine : under which form the prin- 
ciple of vis viva resolves itself into the principle of virtual 
velocities. So many difficulties, however, oppose themselves 
to the introduction of the term vis inertice, associated with 
the definitive idea of a force, into the discussion of questions 
of mechanics, and especially of practical and elementary 
mechanics, that I have thought it desirable to avoid it. It 
is with this view that I have given a new interpretation to 
that function of the velocity of a moving body which is 
known as its vis viva. One half that function I have inter- 
preted to represent the number of units of work accumulated 

* See Poncelet, Mecanique Industricllc, troisieme partie. 



N PR] I 

in the body bo long as its motion is continued. This number 
of units of work it is capable of reproducing upon any resist- 
ance opposed to it> motion. A very simple investigation 
(Art 66.) establishes the truth of this interpretation, and 
3 to the principle of vis viva the following more simple 
enunciation : — "The difference between the aggregate work 
done upon the machine, during any time, by those forces 
which tend to accelerate the motion, and the aggregate 
\\«.rk, during the same time, of those which tend to retard 
the motion, is equal to the aggregate number of units of 
work accumulated in the moving parts of the machine 
during that time if the former aggregate exceed the latter, 
and lost from them during that time if the former aggregate 
fall short of the latter." Tims, then, if the aggregate work 
of the forces which tend to accelerate the motion of a 
machine exceeds that of the forces which tend to retard it, 
then la the surplus work (that done upon the driving points, 
above that expended upon the prejudicial resistances and 
apon the working points) continually accumulated in the 
moving elements of the machine, and their motion is thereby 
continually accelerated. And if the former aggregate be 
less than the latter, then is the deficiency supplied from the 
work already accumulated in the moving elements, so that 
their motion is in this case continually retarded. 

The moving power divides itself whilst it operates in a 
machine, first, into that which overcomes the prejudicial 
itances of the machine, or those which are opposed by 
friction and other causes, uselessly absorbing the work in its 
transmission. Secondly, into that which accelerates the 
motion of the various moving parts of the machine, and which 
accumulates in them so long as the work done by the moving 
power upon it exceeds that expended upon the various 
ances opposed to the motion of the machine. Thirdly, 
into thai which overcomes the useful resistances, or those 
which are opposed to the motion of the machine at the 
working point, or points, by the useful work which is done 
by it. 



PREFACE. XI 

Between these three elements there obtains in every 
machine a mathematical relation, which I have called its 
modulus. The general form of this modulus I have discussed 
in a memoir on the " Theory of Machines " published in the 
Philosophical Transactions for the year 184:1. The deter- 
mination of the particular moduli of those elements of 
machinery which are most commonly in use, is the subject 
of the third part of the following work. From a combination 
of the moduli of any such elements there results at once the 
modulus of the machine compounded of them. 

When a machine has acquired a state of uniform motion, 
work ceases to accumulate in its moving elements, and its 
modulus assumes the form of a direct relation between the 
work done by the motive power upon its driving point and 
that yielded at its working points. I have determined by a 
general method* the modulus in this case, from that statical 
relation between the driving and working pressures upon 
the machine which obtains in the state bordering upon its 
motion, and which may be deduced from the known condi- 
tions of equilibrium and the established laws of friction. In 
making this deduction I have, in every case, availed myself 
of the following principle, first published in my paper on the 
theory of the arch, read before the Cambridge Philosophical 
Society in Dec. 1833, and printed in their Transactions of 
the following year: — "In the state bordering upon motion 
of one body upon the surface of another, the resultant 
pressure upon their common surface of contact is inclined 
to the normal, at an angle whose tangent is equal to the 
coefficient of friction." 

This angle I have called the limiting angle of resistance. 
Its values calculated, in respect to a great variety of surfaces 
of contact, are given in a table at the conclusion of the 
second part, from the admirable experiments of M. Morin,f 
into the mechanical details of which precautions have been 
introduced hitherto unknown to experiments of this class, 

* Art. 152. See Phil. Trans., 1841, p. 290. 

\ Kouvelles Experiences mr le Frottement, Paris, 1833. 



Xll TKl.FACE. 

and which have given to our knowledge of the laws of 
; .»n a precision and a certainty hitherto unhoped for. 

Of the varionfl elements of machinery those which rotate 
indrical axes are of the most frequent occurrence 
and the most useful application; I have, therefore, in the 
first place sought t.> establish the general relation of the 
stat.' bordering opon motion between the driving and the 
working pressnres upon such a machine, reference being 
had to the weight of the machine.* This relation points out 
the < of a particular direction in which the driving 

ue should be applied to any such machine, that the 
amount of work expended upon the friction of the axis may 
be the Least possible. This direction of the driving pressure 
always presents itself on the same side of the axis with that 
of the working pressure, and when the latter is vertical it 
become- parallel to it; a principle of the economy of power 
in machinery which has received its application in the 
parallel motion of the marine engines known as the Gorgon 
Engines. 

I have devoted a considerable space in this portion of my 
work t<> the determination of the modulus of a system of 
toothed wheels ; this determination I have, moreover, 
extended to bevil wheels, and have included in it, with the 
influence of the friction of the teeth of the wheels, that of 
their axes and their weights. An approximate form of this 
modulus applies to any shape of the teeth under which they 
may be made to work correctly; and when in this approxi- 
mate form of the modulus the terms which represent the 
influence of the friction of the axis and the weight of the 
wheel are neglected, it resolves itself into a well known 
theorem of M. Poneelet, reproduced by M. Kavier and the 
Rev. Dr. "WhewelLf In respect to wheels having epicy- 

* In my memoir on the "Theory of Machines" (Phil. Trans. 1841), I have 
extended thi* relation to the case in which the number of the pressures and 
their direct: whatever. The theorem which expresses it is given in 

the Appendix of this work. 

f In tho diBCOaaon of the friction of the teeth of wheels, the direction of the 
mutual pressures of the teeth is determined by a method first applied by me to 



PREFACE. XI 11 

cloidal and involute teeth, the modulus assumes a character 
of mathematical exactitude and precision, and at once 
establishes the conclusion (so often disputed) that the loss of 
power is greater before the teeth pass the line of centres 
than at corresponding points afterwards ; that the contact 
should, nevertheless, in all cases take place partly before 
and partly after the line of centres has been passed. In the 
case of involute teeth, the proportion in which the arc of 
contact should thus be divided by the line of centres is 
determined by a simple formula ; as also are the best 
dimensions of the base of the involute, with a view to the 
most perfect economy of power in the working of the 
wheels. 

The greater portion of the discussions in the third part of 
my work I believe to be new to science. In the fourth part 
I have treated of " the theory of the stability of structures," 
referring its conditions, so far as they are dependent upon 
the rotation of the parts of a structure upon one another, to 
the properties of a certain line which may be conceived to 
traverse every structure, passing through those points in it 
where its surfaces of contact are intersected by the resultant 
pressures upon them. To this line, whose properties I first 
discussed in a memoir upon " the Stability of a System of 
Bodies in Contact," printed in the sixth volume of the Canib. 
Phil. Trans., I have given the name of the line of resist- 
ance ; it differs essentially in its properties from a line 
referred to by preceding writers under the name of the 
curve of equilibrium or the line of pressure. 

The distance of the line of resistance from the extrados of 
a structure, at the point where it most nearly approaches it, 
I have taken as a measure of the stability of a structure,* and 

that purpose in a popular treatise, entitled Mechanics applied to the Arts, 
published in 1834. 

* This idea was suggested to me by a rule for the stability of revetement 
walls attributed to Vauban, to the effect, that the resultant pressure should 
intersect the base of such a wall at a point whose distance from its extrados ie 
£ths the distance between the extrados at the base and the vertical through 
the centre of gravity. 



XIV PREFACE. 

have called it the modulus of stability; conceiving this 
measure of the stability to be of mure obvious and easier 
application than the coefficient of stability used by the 
French writers. 

That structure in respect to every independent element 
of which the modulus of stability is the same, is evidently 
the structure of the greatest stability having a given quantity 
of material employed in its construction ; or of the greatest 
economy of material having a given stability. 

The application of these principles of construction to the 
theory of piers, walls supported by counterforts and shores, 
buttresses, walls supporting the thrust of roofs, and the 
weights of the floors of dwellings, and Gothic structures, 
has suggested to me a class of problems never, I believe, 
before treated mathematically. 

I have applied the well known principle of Coulomb to 
the determination of the pressure of earth upon revetement 
walls, and a modification of that principle, suggested by M. 
Poncelet, to the determination of the resistance opposed to 
the overthrow of a wall backed by earth. This determina- 
tion has an obvious application to the theory of foundations. 

In the application of the principle of Coulomb I have 
availed myself, with great advantage, of the properties of 
the limiting angle of resistance. All my results have thus 
received a new and a simplified form. 

The theory of the arch I have discussed upon principles 
first laid down in my memoir on " the Theory of the Stability 
of a System of Bodies in Contact," before referred to, and 
subsequently in a memoir printed in the "Treatise on 
Bridges " by Professor Hosking and Mr. Hann.* They 
differ essentially from those on which the theory of Coulomb 
is founded ;f when, nevertheless, applied to the case treated 

* I have made extensive use of the memoir above referred to in the following 
work, by the obliging permission of the publisher, Mr. Weale. 

f The theory of Coulomb was unknown to me at the time of the publication 
of my memoirs printed in the Camb. Phil. Trans. For a comparison of the 
two methods see Mr. Hann's treatise. 



PREFACE. XV 

by the French mathematicians, they lead to identical results 
I have inserted at the conclusion of my work the tables of 
the thrust of circular arches, calculated by M. Garidel from 
formulas founded on the theory of Coulomb. 

The fifth part of the work treats of the "strength of 
materials," and applies a new method to the determination 
of the deflexion of a beam under given pressures. 

In the case of a beam loaded uniformly over its whole 
length, and supported at four different points, I have deter- 
mined the several pressures upon the points of support by a 
method applied by M. Kavier to a similar determination in 
respect to a beam loaded at given points.* 

In treating of rupture by elongation I have been led to a 
discussion of the theory of the suspension bridge. This 
question, so complicated when reference is had to the weight 
of the roadway and the weights of the suspending rods, and 
when the suspending chains are assumed to be of uniform 
thickness, becomes comparatively easy when the section of 
the chain is assumed so to vary its dimensions as to be every 
where of the same strength. A suspension bridge thus 
constructed is obviously that which, being of a given 
strength, can be constructed with the least quantity of 
materials ; or, which is of the greatest strength having a 
given quantity of materials used in its construction. f 

The theory of rupture by transverse strain has suggested 
a new class of problems, having reference to the forms of 
girders having wide flanges connected by slender ribs or by 
open frame work : the consideration of their strongest forms 
leads to results of practical importance. 

In discussing the conditions of the strength of breast- 
summers, my attention has been directed to the best positions 
of the columns destined to support them, and to a comparison 



* As in fig. p. 487. of the following work. 

f That particular case of this problem, in which the weights of the suspending 
rods are neglected, has been treated by Mr. Hodgkinson in the fourth vol. of 
Manchester Transactions, with his usual ability. He has not, however, suc- 
ceeded in effecting its complete solution. 



XVI PREFACE. 

of the strength of a beam carrying a uniform load and sup- 
ported freely at its extremities, with that of a beam similarly 
loaded but having its extremities firmly imbedded in 
masonry. 

In treating of the strength of columns I have gladly 
replaced the mathematical speculations upon this subject, 
which are so obviously founded upon false data, by the 
invaluable experimental results of Mr. E. Hodgkinson, 
detailed in his well known paper in the Philosophical 
Transactions for 1S40. 

The sixth and last part of my work treats on " impact ;" 
and the Appendix includes, together with tables of the 
mechanical properties of the materials of construction, the 
angles of rupture and the thrusts of arches, and complete 
elliptic functions, a demonstration of the admirable theorem 
of M. Poncelet for determining an approximate value of the 
square root of the sum or difference of two squares. 

In respect to the following articles of my work I have tc 
acknowledge my obligations to the work of M. Poncelet, 
entitled Mecaniaue Industrielle. The mode of demonstration 
is in some, perhaps, so far varied as that their origin might 
with difficulty be traced ; the principle, however, of each 
demonstration — all that constitutes its novelty or its value — 
belongs to that distinguished author. 

30,* 38, 40, 45, 46, 47, 52, 58, 62, 75, 10S,f 123, 202, 
2674 268 > 269 > 2 ?°> 3 ± 9 > 35 ±> 365 -§ 

* The enunciation only of this theorem is given in the Mec. Ind., 2me partie, 
Art. 38. 

f Some important elements of the demonstration of this theorem are taken 
from the Mec. Lid., Art. 79. 2me partie. The principle of the demonstration 
is not, however, the same as in that work. 

+ In this and the three following articles I have developed the theory of the 
By-wheel, under a different form from that adopted by M. Poncelet {Mec. Ind., 
A.rt. 56. 3me partie). The principle of the whole calculation is, however, 
taken from his work. It probably constitutes one of the most valuable of his 
contributions to practical science. 

§ The idea of determining the work necessary to produce a given deflection 
of a beam from that expended the compression and the elongation of its com- 
ponent fibres was suggested by an observation in the Mec. Ind., Art. 75. 3me 
partie. 



CONTENTS 



PART I, 



STATICS. 

The Parallelogram of Pressures 

The Principle of the Equality of Moments 

The Polygon of Pressures 

The Parallelopipedon of Pressures 

Of Parallel Pressures 

The Centre of Gravity 

The Properties of Guldinus 



Pago 
3 
6 
10 
14 
16 
20 
86 



PART II. 



DYNAMICS. 



Motion . 47 

Velocity . 48 

Work 48 

Work of Pressures applied in different Directions to a Body moveable 

about a fixed Axis 67 

Accumulation of Work 58 

Angular Velocity 65 

The Moment of Inertia 70 

The Acceleration of Motion by given moving Forces . . .79 

The Descent of a Body upon a Curve 83 

The Simple Pendulum 85 

Impulsive Force ....86 

The Parallelogram of Motion ......... 86 

The Polygon of Motion 88 

The Principle of D'Alembert 89 

Motion of Translation 90 

Motion of Rotation about a fixed Axis 91 

The Centre of Percussion . 96 

The Centre of Oscillation , . 96 

Projectiles 99 

Centrifugal Force 106 

* 



Will CONTENTS. 

Paps 

Tin Principle of virtual Velocities 113 

Principle o\ Vis \ 115 

Dynamic*] Stability 121 

124 

Summary of the Laws of Friction 130 

The limit iii^r Angle of Resistance 181 

B sistencc 133 

The f. rderiug upon Motion 133 

Tns Riqiditt of Cords 142 



PART III. 

THE THEORY OF MACHINES. 

The Transmission of Work by Machines . 146 

The Modulus of a Machine moving with a uniform or periodical Motion . 148 
The Modulus of a Machine moving with an accelerated or a retarded 

Motion 150 

The Velocity of a Machine moving with a variable Motion . . .151 
To determine the Co-efficients of the Modulus of a Machine . . . 153 
General Condition of the State bordering upon Motion in a Body acted 
upon by Pressures in the same Plane, and moveable about a cylindrical 

Axis 154 

The Wheel and Axle . .155 

The Pulley 160 

i of one fixed and one moveable Pulley 161 

\ - stem of one fixed and any Xumber of moveable Pulleys . . . 163 

A Tackle of any Number of Sheaves 166 

The Modulus of a compound Machine 169 

The Capstan 194 

The Chinese Capstan 199 

The Horse Cap-tan, or the Whim Gin 802 

The Friction of Cords 207 

The Friction Break 213 

land 215 

The modulus of the Band 217 

The Teeth of Wheels 227 

Involute Teeth 23 1 

■loidal and Hvpocyeloidal Teeth 2Sfi 

. out the Teeth of Wheels 239 

A Train of Wheels 241 

The Btrength of Teeth 243 

To describe Bptoycloidal Teeth 245 

involute Teeth 251 

The T . ,'id Pinion 253 



CONTENTS. 



XIX 



Pago 
The Teefh of a Wheel working with a Lantern or Trundle . . . 257 

The driving and working Pressures on Spur Wheels 259 

The Modulus of a System of two Spur Wheels 268 

The Modulus of a Rack and Pinion 282 

Conical or Bevil Wheels 284 

The Modulus of a System of two Bevil Wheels 288 

The Modulus of a Train of Wheels 301 

The Train of least Resistance 310 

The Inclined Plane 312 

The Wedge driven by Pressure 321 

The Wedge driven by Impact 323 

The mean Pressure of Impact 325 

The Screw 326 

Applications of the Screw 329 

The Differential Screw 331 

Hunter's Screw 332 

The Theory of the Screw with a Square Thread in reference to the vari- 
able Inclination of the Thread at different Distances from the Axis . 333 

The Beam of the Steam Engine 337 

The Crank 341 

The Dead Points in the Crank 34.5 

The Double Crank 346 

The Crank Guide 351 

The Fly-wheel 353 

The Friction of the Fly-wheel 362 

The Modulus of the Crank and Fly-wheel 363 

The Governor 364 

The Carriage-wheel 368 

On the State of the accelerated or retarded Motion of a Machine . .373 

PART IV. 

THE THEORY OF THE STABILITY OF STRUCTURES. 



General Conditions of the Stability of a Structure of Uncemented Stones 37/ 

The Line of Resistance . . • 377 

The Line of Pressure 379 

The Stability of a Solid Body 880 

The Stability of a Structure . . . 381 

The Wall or Pier 882 

The Line of Resistance in a Pier 383 

The Stability of a Wall supported by Shores 881 

The Gothic Buttress 89C 

The Stability of Walls sustaining Roofs 397 

The Plate Band 402 

The sloping Buttress 



XX 

Pag* 

ibility of a Wall sustaining the Pressure of a Fluid . . . 408 

Earth Works 412 

Mi.nt Walk 416 

Lrch 429 

f Rupture 437 

The I. .in a circular arch whose Youssoirs are equal, and 

I distributed orer different Points of its Extrados . . 44 < > 

entel Arch irhoae Extrados is horizontal 441 

a ' the Extrados of each Semi-Arch being a straight Line 

inclined at any given Angle to the Horizon, and the Material of the 

I ling different from that of the Arch 442 

A circular Arch having equal Youssoirs and sustaining the Pressure of 

Water 444 

The Equilibrium ol' an Arch, the Contact of whose Youssoirs is geometri- 
cally accurate 446 

Applications of the Theory of the Arch 448 

Tables of the Thrust of Arches 454 



PART Y. 

THE STRENGTH OF MATERIALS. 

Elasticity 458 

Elongation 459 

The Moduli of Resilience and Fragility 462 

Deflection 467 

The Deflexion of Beams loaded uniformly 481 

The Deflexion of Breast Summers 4S6 

Rupture 509 

Tenacity BOS 

The Suspension Bridge 

The Catenary 5uC 

-on Bridge of greatest Strength 510 

Rupture by Compression 51 S 

tion of Rupture in a Beam 

al Conditions of the Rupture of a Beam 521 

The Beam Strength 527 

The Strength off Breast Summers 540 

Tin- beat Positions of their Points of Support 542 

Forinuhe representing the absolute Strength of a Cylindrical Column to 

sustain ■ Pressure in the Direction of its Length 545 

Torsion 546 



CONTENTS. 



XXI 



PART VI. 

IMPACT. 

Page 
The Impact of two Bodies whose centres of Gravity move in the same 

right Line 553 

Greatest Compression of the Surface of the Bodies 555 

Velocity of two elastic Bodies after Impact 556 

The Pile Driver 564 

Additions by the American Editor 571 



APPENDIX. 



subject to 



Note A 

Note B. — Poncelet's Theorems . 

Note C— On the Rolling of Ships . 

Note D 

Note E. — On the Rolling Motion of a Cylinder 

Note F. — On the Descent upon an Inclined Plane of a Bod 
Variations of Temperature, and on the Motion of Glaciers 

Note G. — The best Dimensions of a Buttress 

Note H. — Dimensions of the Teeth of "Wheels . 

Note I. — Experiments of M. Morin on the Traction of Carriages 

Note K. — On the Strength of Columns 

Table I. — The Numerical Values of complete Elliptic Functions of the 
first and second Orders for Values of the Modulus Tc corresponding to 
each Degree of the Angle sin.- l k 

Table II. — Showing the Angle of Rupture -f of an Arch whose Loading 
is of the same Material with its Voussoirs, and whose Extrados is 
inclined at a given Angle to the Horizon 

Table III. — Showing the Horizontal Thrust of an Arch, the Radius of 
whose Intrados is Unity, and the Weight of each Cubic Foot of its 
Material and that of its Loading, Unity 

Table IV. — Mechanical Properties of the Materials of Construction . 

Table V.— Useful Numbers 



631 
632 

637 
653 
655 

675 
683 
684 
6S5 
686 



687 



688 



691 
694 
698 



THE 



MECHANICAL PRINCIPLES 



OF 



CIVIL ENGINEERING 



FA.RT I 

STATICS. 



1. Force is that which tends to cause or to destroy 
motion, or which actually causes or destroys it. 

The direction of a force is that straight line in which it 
tends to cause motion in the point to which it is applied, or 
in which it tends to destroy the motion in it.* 

When more forces than one are applied to a body, and 
their respective tendencies to communicate motion to it 
counteract one another, so that the body remains at rest, 
these forces are said to be in equilibrium, and are called 

PRESSURES. 

It is found by experiment f that the effect of a pressure, 
when applied to a solid body, is the same at whatever point 
in the line of its direction it is applied ; so that the condi- 
tions of the equilibrium of that pressure, in respect to other 
pressures applied to the same body, are not altered, if, with 
out altering the direction of the pressure, we remove its 
point of application, provided only the point to which we 
remove it be in the straight line in the direction of which it 
acts. 

The science of Statics is that which treats of the equili- 
brium of pressures. When two pressures only are applied to 

* Note (a) Ed. Appendix. \ Note (b) Ed. Appendix. 



2 THE I'M'i' OF PBE88UBE. 

r body, ana bold it al rest, it is found by experiment that 
these pressures acl In opposite directions, and have their 
directions always in the same straight line. Two such pres- 
snros are said to be equal. 

\\\ instead of applying two pressures which are thus equal 
in opposite directions, we apply them both in the same 
direction, the .-ingle pressure which must be applied in a 
direction opposite to the two to sustain them, is said to be 
double of either of them. If we take a third pressure, equal 
to either of the two first, and apply the three in the same 
direction, the single pressure, which must be applied in a 
direction opposite to the three to sustain them, is said to be 
triple of cither of them : and so of any number of pressures. 
Thus, fixing upon any one pressure, and ascertaining how 
many pressures equal to this are necessary, when applied in 
an opposite direction, to sustain any other greater pressure, 
we arrive at a true conception of the amount of that greater 
pressure in terms of the first. 

That single pressure, in terms of which the amount of any 
other greater pressure is thus ascertained, is called an unit 
of pressure. 

Pressures, the amount of which are determined in terms 
of some known unit of pressure, are said to be measured. 

Different pressures, the amounts of w r hich can be deter- 
mined in terms of the same unit, are said to be commensur- 
able. 

The units of pressure which it is found most convenient to 
use, are the weights of certain portions of matter, or the 
pressures with which they tend towards the centre of the 
earth. The units of pressure are different in different coun- 
tries. "With us, the unit of pressure from which all the rest 
are derived is the weight of 22*815* cubic inches of distilled 
water. This weight is one pound troy; being divided into 
5760 equal parts, the weight of each is a grain troy, and 
7000 such grains constitute the pound avoirdupois. 

If straight lines be taken in the directions of any number 
of pressures, and have their lengths proportional to the 
numbers of units in those pressures respectively, then these 
lines having to one another the same proportion in length 
that the pressures have in magnitude, and being moreover 
drawn in the directions in which those pressures respectively 
act, are said to /< present them in magnitude and direction. 

* This standard was fixed bj Act of Parliament, in 1824. The temperature 
of the water i> supposed to be 62 c Fahrenheit, the weight to be taken in air, 
and the barometer to stand at 30 inches. 



THE PARALLELOGRAM OF PRESSURES. 3 

A system of pressures being in equilibrium, let any num- 
ber of them be imagined to be taken away and replaced by 
a single pressure, and let this single pressure be such that 
the equilibrium which before existed may remain, then this 
single pressure, producing the same effect in respect to the 
equilibrium that the pressures which it replaces produced, is 
said to be the resultant. 

The pressures which it replaces are said to be the compo- 
nents of this single pressure ; and the act of replacing them 
by such a single pressure, is called the composition of 
pressures. 

If, a single pressure being removed from a system in equi- 
librium, it be replaced by any number of other pressures, 
such, that whatever effect was produced by that which they 
replace singly, the same effect (in respect to the conditions of 
the equilibrium) may be produced by those pressures con- 
jointly, then is that single pressure said to have been re- 
solved into these, and the act of making this substitution 
of two or more pressures for one, is called the resolution 
of pressures. 

The Parallelogram of Pressures. 

2. The resultant of any two pressures applied to a point, 
is represented in direction by the diagonal of a paral- 
lelogram, whose adjacent sides represent those pressures in 
magnitude and direction* 

(Duchayla's Method.f) 

To the demonstration of this proposition, after the excel- 
lent method of Duchayla, it is necessary in the first place 
to show, that if there be any two pressures P 2 and P 3 whose 
directions are in the same straight line, and a third pressure 
P, in any other direction, and if the proposition be true in 
respect to P, and P 2 , and also in respect to P x and P 3 , then 
it will be true in respect to P, and P 2 + P 3 . 

Let P 1? P 2 , and P 3 , form part of any system of pressures in 
.«.,*? equilibrium, and let them be applied to the point 

\V.. % *T\\ A; take AB and AC to represent, in magnitude 

\ m **5^4 anc * direction, the pressures P, and P 5 , and CD 

> -*->-* the pressure P 3 , and complete the parallelograms 

CB and DF. Suppose the proposition to be true with regard 

* This proposition constitutes the foundation of the entire science of Statics. 
f Note (c) Ed. App. 



4 THE r.\K.\i.i.] i.(k.i:am 

to IV and I\. the reeultanl of P, Bnd P, will then be in the- 
direction of the diagonal AE of the parallelogram P>( I, whose 
adjacent Bidee AC and All represent P, ami r, in magnitude 
and direction. Let P, and P, be replaced bv this resultant. 
It matters not to the equilibrium where in tne line AP it is 
applied; Let it then be applied at 1". But thus applied at 
Kir may, without affecting the conditions of the equilibrium, 
be in its turn replaced by (or resolved into) two other pressures 
acting in OF and BF, and these will manifestly be equal to 
P, and P t , of which P, may be transferred without altering 
the conditions to C, and P 2 to E. Let this be done, and let 
P 8 be transferred from A to C, we shall then have P x and 
P, acting in the directions CF and CD at C and P 2 , in the 
direction IT. at E, and the conditions of the equilibrium will 
not have been affected by the transfer of them to these 
points. .Now suppose that the proposition is also true in 
respect to P, and P, as well as P, and P 2 . Then since CF 
and CD represent P, and P 3 in magnitude and direction, 
therefore their resultant is in the direction of the diagonal 
GE. Let them be replaced by this resultant, and let it be 
transferred to E, and let it then be resolved into two other 
pressures acting in the directions DE and FE ; these will 
evidently be P, and P 3 . We have now then transferred all 
the three pressures P 15 P 2 , P 8 , from A to E, and they act at E 
in directions parallel to the directions in which they acted at 
A, and this has been done without affecting the conditions of 
the equilibrium; or, in other words, it has been shown that 
the pressures P„ P 2 , P s , produce the same effect as it re- 
spects the conditions of the equilibrium, whether they be 
applied at A or E. The resultant of P„ P 2 , P 3 , must there- 
fore produce the same effect as it regards the conditions of 
the equilibrium, whether it be applied at A or E. But in 
order that this resultant may thus produce the same effect 
when acting at A or E, it must act in the straight line AE, 
because a pressure produces the same effect when applied at 
two different points only when both those points are in the 
line of its direction. On the supposition made, therefore, 
the resultant of P„ P„ and P„ or of P, and P Q + P 3 
acts in the direction of the diagonal AE of the parallel- 
ogram I'D, whose adjacent sides AD and AJ3 represent 
P a + P, and Pj in magnitude and direction; audit has been 
shown, that if the proposition be true in respect to P t and 
P„ and also in respeel to P, and P„ then it is true in respect 
to P, and P, + l\. Now this being the case for all values 
of P„ I\, P 3 , it is the case when P„ P 9 , and P 8 , are equal 



OF PRESSURES. 5 

to one another. But if P x be equal to P 2 their resultant 
will manifestly have its direction as much towards one of 
these pressures as the other ; that is, it will have its direc- 
tion midway between them, and it will bisect the angle BAC : 
but the diagonal AF in this case also bisects the angle BAC, 
since P x being equal to P 2 , AC is equal to AB ; so that in 
this particular case the direction of the resultant is the 
direction of the diagonal, and the proposition is true, and 
similarly it is true of P x and P 3 , since these pressures are 
equal. Since then it is true of P x and P 2 when they are 
equal, and also of P x and P 3 , therefore it is true in this case 
of P a and P 2 + P 3 , that is of P, and 2 P r And since it is 
true of P l and P 2 , and also of P x and 2 P n therefore it is true 
of Pj and P 2 + 2 P l5 that is of P x and 3 P x ; and so of P x and 
m Pj, if m be any whole number ; and similarly since it is 
true of m P a and P„ therefore it is true of m P, and 2 P n &c, 
and of m P a and n P, where n is any whole number. There- 
fore the proposition is true of any two pressures m P x and 
n Pj which are commensurable. 

It is moreover true when the pressures are in- 
c t%- ;''.;.'"/ commensurable. For let AC and AB represent 
•f:.:::::J:Bg a ny two such pressures P x and P 2 in magnitude 
and direction, and complete the parallelogram 
ABDC, then will the direction of the resultant of P a and 
P„ be in AD ; for if not, let its direction be AE, and draw 
EG parallel to CD. Divide AB into equal parts, each less 
than GO, and set off on AC parts equal to those from A 
towards 0. One of ih.Q divisions of these will manifestly 
fall in GO. Let it be H, and complete the parallelogram 
.VII FB. Then the pressure P 2 being conceived to be 
divided into as many equal units of pressure as there are 
equal parts in the line AB, All may be taken to represent a 
pressure P 3 containing as many of these units of pressure 
as there are equal parts in AH, and these pressures P 2 and 
P., will be commensurable, being measured in terms of the 
same unit. Their resultant is therefore in the direction AF, 
and this resultant of P 3 and P 2 has its direction nearer to 
AC than the resultant AE of P x and P 2 has ; which is 
absurd, since P, is greater than P 3 . 

Therefore AE is not in the direction of the resultant of 
Pj and P 2 ; and in the same manner it may be shown that no 
other than AD is in that direction. Therefore, &c. 



-c 



THE PBINCTPLE8 OF THE 



3. The resultant of two pressures applied in any directions 
to (i pointy is n presented in magnitude as well as in direc- 
tion oy tin </i<!(jo7ial of the parallelogram whose adjacent 
sides represent those pressures in magnitude and in direc- 
tion. 

Let BA and CA represent, in magnitude and 
/\ direction, any two pressures applied to the point 

A. C< >n i plete the parallelogram BC. Then by 
the last proposition AD will represent the result- 
ant of these pressures in direction. It will also 
represent it in magnitude ; for, produce DA to G, and con- 
ceive a pressure to be applied in GA equal to the resultant 
of BA and CA, and opposite to it, and let this pressure be 
represented in magnitude by the line GA. Then will the 
pressures represented by the lines BA, CA, and GA, mani- 
festly be pressures in equilibrium. Complete the parallelo- 
gram BG, then is the resultant of GA and BA in the 
direction FA ; also since GA and BA are in equilibrium 
with CA, therefore this resultant is in equilibrium with CA, 
but when two pressures are in equilibrium, their directions 
are in the same straight line ; therefore FAC is a straight 
line. But AC is parallel to BD, therefore FA is parallel to 
BD, and FB is, by construction, parallel to GD, therefore 
AFBD is a parallelogram, and AD is equal to FB and 
therefore to AG. But AG represents the resultant of CA 
and BA in magnitude, AD therefore represents it in magni- 
tude. Therefore, &c* 



The Principle of the Equality of Moments. 

4. Definition. If any number of pressures act in the 
same plane, and any point be taken in that plane, and per- 
pendiculars be drawn from it upon the directions of all these 
pressures, produced if necessary, and if the number of units 
in each pressure be then multiplied by the number of units 
in the corresponding perpendicular, then this product is 
called the moment of that pressure about the point from 
which the perpendiculars are drawn, and these moments are 
said to be measured from that point. 

* Note (d) Ed. App. 



EQUALITY OF MOMENTS. 



5. If three pressures he in equilibrium, and their moments 
be taken about any point in the plane in which they act, 
then the sum of the moments of those two pressures which 
tend to turn the plane in one direction about the point 
from which the moments are measured, is equal to the 
moment of that pressure which tends to turn it in the 
opposite direction. 

f^^. 3.....x.je ^ e t Pi? P a ? P35 acting in the directions 

° ;p t"-X>^ ;/\ I\0, P 2 0, P 3 0, be any three pressures in 
D-,.„..^^B equilibrium. Take any point A in the plane 

. »■""" in which they act, and measure their moments 

from A, then will the sum of the moments of P 2 and P 3 , 
which tend to turn the plane in one direction about A, equal 
the moment of P 1? which tends to turn it in the opposite 
direction. 

Through A draw DAB parallel to OP l5 and produce OP 2 
to meet it in D. Take OD to represent P 2 , and take DB 
such a length that OD may have the same proportion to 
DB that P 2 has to P r Complete the parallelogram ODBC, 
then will OD and OC represent P 2 and P x in magnitude and 
direction. Therefore OB will represent P 3 in magnitude 
and direction. 

Draw AM, AN, AL, perpendiculars on OC, OD, OB, 
and join AO, AC. Now the triangle OBC is equal to the 
triangle OAC, since these triangles are upon the same base 
and between the same parallels. 

Also, A ODA+AOAB = AOBD = AOBC, _ v 

.-.A ODA+AOAB = A OAC, »>^'~- -f^ 

.-. j OD x AN+j OB x AL = j OC x AM, X^xft], 

. • . P a x AN+P 3 x AL=P, x AM. *" — * ' 

Now P x x AM, P 2 x AN, P 3 x AL, are the moments of P 1? 
P 2 , P 3 , about A (Art. 4.) 

.•.m t P a + m t P 3 = m t P 1 (1). 

Therefore, &c. &c. 

6. If R be the resultant of P 2 and P 3 , then since R is 
equal to P x and acts in the same straight line, lmR = mtP,, 
.•.m t P a + m t P 3 = m t R. 

The sum of the moments therefore, about any point, of 
two pressures, P a and P 3 in the same plane, which tend to 



5 THE PE] «)F THE 

turn it in the >ame direction about that point, is equal tc 
the moment of their resultanl about that point. 

If they had tended to turn it in opposite directions, then 
the different of their moments would have equalled the 
moment of their resultant. For let II be the resultant of 
P, and P l3 which tend to turn the plane in opposite direc- 
tion- about A. &c. Then is R equal to P 9 , and in the same 
straight line with it, therefore moment R is equal to 
moment P„. But by equation (1) mT, — mVP, = mtP 9 ; 
.'.mT,- m t P, = m t K. 

Generally, therefore, m* P t + m* P, = m* R (2), 

the mown nb, therefore, of the resultant of any two pressures 
in the same plane is equal to the sum or difference of the 
moments of its components, according as they act to turn the 
phi ue in the same direction about the point from which the 
moments are measured, or in opposite directions:' 



vr 



7. If any number of pressures in the same plane be in equi- 
Ubrivm* and any point be taken, in that plane, from 
which their moments are measured, then the sum of the 
moments of those pressures which tend to turn the plane 
in one direction about that point is equal to the sum of the 
moments of those which tend to turn it in the opposite 
direction. 

Let P 1? P 2 , P 3 P» be any number of pressures in 

the same plane which are in equi- 

p " jL * , librium. and A any point in the 

■ ^f y ^ ^^^ pl an e from which their moments 

*S 'j£v 5 ? are measured, then will the sum of 

l£ the moments of those pressures 

which tend to turn the plane in one direction about A equal 

the Bum of the moments of those which tend to turn it in 

the opposite direction. 

Let R, be the resultant of P 1 and P 2 , 

R, R, and P 3 , 

R 3 R 3 and P 4 , 

&c &c. 

Rn-i R«-2 and P n . 

Therefore, by the Last proposition, it beine: understood 
that the moments of those oi the pressures r^ P 2 , which 
tend to turn the plane to the left of A, are to be taken nega- 
tively, we have 

* Note (c) Ed. App. 



EQUALITY OF MOMENTS, V 

m* K x = in* P x + m* P t . 

m* E 2 = m* R x + m* P 3 , 

m* R 3 — m* E 2 + m* P 4 , 

&c. = &c. &c. 

m* E„_i = m* Un-2 + m* P w . 

Adding these equations together, and striking ont the 
terms common to both sides, we have 

m* R„_i = in* P x + m* P 2 + m* P 3 + + m* P„ 

. . . (3), where Pn_i is the resultant of all the pressures P 1? 
P P 

But these pressures are in equilibrium ; they have, there- 
fore, no resultant. 

.-. IV-i = .-. m* R„_i = 0, 
. •. m* P, + m* P 2 + m* P 3 , + m* P w = . . . . (4). 

Now, in this equation the moments of those pressures which 
tend to turn the system to the left hand are to be taken 
negatively. Moreover, the sum of the negative terms must 
equal the sum of the positive terms, otherwise the whole 
sum could not equal zero. It follows, therefore, that the 
sum of the moments of those pressures which tend to turn 
the system to the right must equal the sum of the moments 
of those which tend to turn it to the left. Therefore, &c. &c. 



8. If any number of pressures acting in the same plane be in 
equilibrium, and they be imagined to be moved parallel to 
their existing directions, and all applied to the same point, 
so as all to act upon that point in directions parallel to 
those in which they before acted upon different points, then 
will they be in equilibrium about that point. 

For (see the preceding figure) the pressure R, at whatever 
point in its direction it be conceived to be applied, may be 
resolved at that point into two pressures parallel and equal 
to Pi and P 2 : similarly, P 2 may be resolved, at any point in 
its direction, into two pressures parallel and equal to R x and 
P 3 , of which Rj may be resolved into two, parallel and equal 
to Pj and P 2 , so that P 2 may be resolved at any point of its 
direction into three pressures parallel and equal to P l5 P 2 , P 3 : 
and, in like manner, P 3 may be resolved into two pressures 
parallel and equal to R 2 and P 4 , and therefore into four pres- 
sures parallel and equal to P„ P 2 , P 3 , P 4 , and so of the rest 



10 THE POLYGON 

Therefore K, .1 maw at any point of its direction be resolved 

in!., n pressures parallel and equal to P„ P„ P 3 , P n ; 

it', therefore, n such pressures were applied to that point, 
they would just be held in equilibrium by a pressure equal 
and opposite to R»_ 1. But R«_i = 0; these n pressures 
would, therefore, be in equilibrium with one another if 
applied to this point. 

Now it is evident, that if, being thus applied to this point, 
they would be in equilibrium, they would be in equilibrium 
if similarly applied to any other point. Therefore, &c. 



The Polygon of Pressures. 

The conditions of the equilibrium of any number of pres- 
sures applied to a point. 

Let OP„ OP 2 , OP 3 , &c, represent in mag- 
nitude and direction pressures P„ P 2 , &c, 
applied to the same point O. Complete the 
parallelogram OP, AP 9 , and draw its diago- 
nal OA ; then will OA represent in magni- 
tude and direction the resultant of P, and 
P a . Complete the parallelogram OABP,, then will OB 
represent in magnitude and direction the resultant of OA 
and P 3 ; but OA is the resultant of P, and P 2 , therefore OB 
is the resultant of P„ P 2 , P 3 ; similarly, if the parallelogram 
OBCP 4 be completed, its diagonal 0(5 represents the result- 
ant of OB and P 4 , that is, of P„ P 2 , P 9 , P 4 , and in like 
manner OD, the diagonal of the parallelogram OCDP 5 , 
represents the resultant of P„ P 2 , P 3 , P 4 , P B . 

Now let it be observed, that AP, is equal and parallel to 
OP 2 , AB to OP 3 , BC to OP 4 , CD to OP 5 , so that P,A, AB, 
BC, CD, represent P„, P 3 , P 4 , P 6 , respectively in magnitude, 
and are parallel to their directions. Moreover, OP, is in the 
direction of P, and represents it in magnitude, so that the 
sides OP,. P,A, AB, BC, CD, of the polygon OP„ ABCDO 
represent the pressures P„ P 2 , P 8 , P 4 , P 5 , respectively in 
magnitude, and are parallel to their directions; whilst the 
Bide ()1), which completes that polygon, represents the 
resultant of those pressures in magnitude and direction. 

If, therefore, the pressures P„ P„ P 3 , P 4 , P 6 , be in equili- 
brium, so that they have no resultant, then the side OD of 
the polygon must vanish, and the point D coincide with O. 
Thus, then, if any number of pressures be applied to a point 




OF PRESSURES. 11 

and lines be drawn parallel to the directions of those pres- 
sures, and representing them in magnitude, so as to form 
sides of a polygon (care being taken to draw each line from 
the point where it unites with the preceding, towards the 
direction in which the corresponding pressure acts), then the 
line thus drawn parallel to the last pressure, and representing 
it in magnitude, will pass through the point from which the 
polygon commenced, and will just complete it if the pres- 
sures be in equilibrium ; and if they be not in equilibrium, 
then this last line will not complete the polygon, and if a 
line be drawn completing it, that line will represent the 
resultant of all the pressures in magnitude and direction. 

This principle is that of the polygon of pressures ; it 
obtains in respect to pressures applied to the same point, 
whether they be in the same plane or not. 

10. If any number of pressures in the same plane he in equi- 
librium, and each be resolved in directions parallel to any 
two rectangular axes, then the sum of all those resolved 
pressures, whose tendency is to communicate motion in one 
direction along either axis, is equal to the sum of those 
whose tendency is in the opposite direction. 

Let the polygon of pressures be formed in respect to any 
number of pressures, P 15 P 2 , P 3 , P 4 , in the same plane and in 
equilibrium (Arts. 8, 9), and let the sides of 
this polygon be projected on any straight line 
Ax in the same plane. JSTow it is evident, 
that the sum of the projections of those sides 
---ce of the polygon which form that side of the 
figure which is nearest to Ax, is equal to the sum of the pro- 
jections of those sides which form the opposite side of the 
polygon : moreover, that the former are those sides of the 
polygon which represent pressures tending to communicate 
motion from A towards x, or from left to right in respect to 
the line Ax / and the latter, those which tend to communi- 
cate motion in the opposite direction. Now each projection 
is equal to the corresponding side of the polygon, multiplied 
by the cosine of its inclination to Ax. The sum of all those 
sides of the polygon which represent pressures tending to 
communicate motion from A towards x, multiplied each by 
the cosine of its inclination to Ax, is equal, therefore, to the 
sum of all the sides representing pressures whose tendency 
is in the opposite direction, each being similarly multiplied 
by the cosine of its inclination to Ax. Now the sides of the 




12 Nil. £] 30L1 

polygon represenl the pressures in magnitude, and are 
inclined al the Bame angles A . Therefore, each pressure 
g multiplied by the cosine of its inclination to Aa*, the 
Bum of all these products, in respect to those which tend to 
communicate motion in one direction, equals the Bum simi- 
larly taken in I to those which tend to communicate 
motion in the opposite direction; or, if in taking this sum it 
be unders ood tnal each term into which there enters a pres- 
Bure, whose tendency is from A towards a?, is to be taken 
positively, whilst each into which there enters a pressure 
which tends from ./' towards A is to be taken negatively, 
then the Bum of all these terms will equal zero; that is, 
calling the inclinations of the directions of P l5 P 2 , P, . . . P„ 
to A?, a 1? a ? , a 3 . . . . a n respectively, 

P t cos. a l -f P 2 cos.a 3 + P 3 cos.a 3 + + P» cos. a n =0 . . . (5), 

in which expression all those terms are to be taken nega- 
tively which include pressures, whose tendency is from x 
toward- A. 

This proposition being true in respect to any axis, Ass is 
true in respect to another axis, to which the inclinations of 
the direction- of the pressures are represented by j3 l? &, (3„ 
fin , so that, 

P, cos. & x + P, cos. 2 + . . . . + P n cos. p n =0. 

Let this second axis be at right angles to the first : 

then 0, — — — a x .*. cos. j3 x = sin. o 1? ft= — — a 2 , ,\ cos. j3 2 

= sin. a„ &c. = &c. 

.-. P x sin. a x + P 2 sin. a % + -f P n sin. a n = ' (6) ; 

those terms in this equation, involving pressures which tend 
to communicate motion in one direction, in respect to the 
axis Ay being taken with the positive sign, and those which 
tend in the opposite direction with the negative sign. 

W the pressures P„ P„ &c. be each of them resolved 
into two others, one of which is parallel to the axis Ax, and 
the other to the axis Ay. it is evident that the pressures 
thus resolved parallel to As. will be represented by r,cos. a„ 
P 3 cos. </.. &c, and those resolved parallel to Ay, by 
in. o, P, Bin. «,, &c Thus then it follows, that if 
any svstem of pressures in equilibrium be thus resolved 
parallel to two rectangular axes, the sum of those resolved 
jsures, whose tendency is in one direction along either 



OF PRESSTTEES. 13 

axis, is equal to the sum of those whose tendency is in the 
opposite direction.* 

This condition, and that of the equality of moments, are 
necessary to the equilibrium of any number of pressures in 
the same plane, and they are together sufficient to that equi- 
librium. 

11. To determine the resultant of any number of pressures 
in the same plane. 

? If the pressures P x P 2 .... P w be not in 

I Jf ■-— * 3 equilibrium, and have a resultant, then one 

*\ ; i \ Pj side is wanting to complete the polygon of 

j :%>p^ ! pressures, and that side represents the res- 
A i — ±JL.j__.LLl^.. ultant of all the pressures in magnitude, 
and is parallel to its direction (Art. 9). 
Moreover it is evident, that in this case the sum of the pro- 
jections on Ax (Art. 10) of those lines which form one 
side of the polygon, will be deficient of the sum of those of 
the lines which form the other side of the polygon, by the 
projection of this last deficient side ; and therefore, that the 
sum of the resolved pressures acting in one direction along 
the line Ax, will be less than the sum of the resolved pres- 
sures in the opposite direction, by the resolved part of the 
resultant along this line. E"ow if R represent this resultant, 
and 6 its inclination to Ax, then R cos. is the resolved part 
of R in the direction of Ax. Therefore the signs of the terms 
being understood as before, we have 

R cos. 0=T X cos. «j + P 2 cos. a 2 -f . . . . +P n cos. a n . . (7). 

And reasoning similarly in respect to the axis Ay, we have 

R sin. 0=Pj sin. a i -fP Q sin. a 2 + .... +P w sin. a n . . . (8). 

Squaring these equations and adding them, and observing 
that R 2 sin. 2 + R 2 cos. 2 0=R 2 (sin. 2 0+cos. 2 6>) =R 2 , we have 

R 2 =(HP sin. a) 2 + (2P cos. a) 2 (9), 

where 2P sin. a is taken to represent the sum P, sin. a, -i- 
P 2 sin. a a + P 3 sin. a 3 + &c, and 2P cos. a to represent the 
sum P, cos. a, + P 2 cos. « 2 + P 3 cos. a 3 + &c. 
Dividing equation (8) by equation (7), 

tan.0=5L5L!L? (10). 

2P cos. a v J 

Thus then by equation (9) the magnitude of the resultant 
* Note (/ ) Ed. App. 



14: THE PARALLKLOPIPEDON 

R is known, and by equation (10) its inclination 0to the axis 
A./* ifl known, [n order completely to determine it, we have 
yet to find the perpendicular distance at which it acts from 
the giveo point A. For this we must have recourse to the 
condition of the equality of moments I Art. 7). 

It' the Bum of the moments of those of the pressures, P„ 
P, . . . . P», which tend to turn the system in one direc- 
tion about A, do not equal the sum of the moments of those 
which tend to turn it the other way, then a pressure being 
applied to the system, equal and opposite to the resultant R, 
will bring about the equality of these two sums, so that the 
moment of R must be equal to the difference of these sums. 
Let then j> equal the perpendicular distance of the direction 
of R from A. Therefore 

R^=m t P 1 + in t P a + m t P I + .... +m t P n . . . (11), 

in the second member of which equation the moments of 
those pressures are to be taken negatively, which tend to 
communicate motion round A towards the left. 
Dividing both sides by R we have 

m t P, + m t P,+ .... + m'P, ( 

■P~ R K >' 

Thus then by equations (9), (10), (12), the magnitude of 
the resultant R, its inclination to the given axis Ax, and the 
perpendicular distance of its direction from the point A, are 
known; and thus the resultant pressure is completely deter- 
mined in magnitude and direction. 

The Parallelopipedon of Pres stores. 

12. Three pressures, P l? P„ P 3 , being applied to the sam£ 
point A, in directions xA, yA, zA, which are not in the 
same plane, it is required to determine their residtant. 

Take the lines P, A, P 2 A, P 3 A, to represent the pressures 

P„ P a , P 3 , in magnitude and direction. 

J Complete the parallelopipedon RP^PjP,, 

/f\T2t^ ot which AP n AP 2 , AP 3 , are ad j acent edges, 

f~Y~/Hi an( l draw it's diagonal RA ; then will RA 

l fK/i /■**'"* represent the resultant of P„ P 2 , P ? , in 

Jv : s' direction and magnitude. For since 

P,SP 9 A is a parallelogram, whose adjacent 

sides P, A, P 9 A, represent the presurea 

P, and P a in magnitude and direction, therefore its diagonal 



OF THREE PRESSURES. 15 

SA represents the resultant of these two pressures. And 
similarly HA, the diagonal of the parallelogram RSAP 3 , re- 
presents in magnitude and direction the resultant of S A and 
P 3 , that is, of P l5 P 2 and P 3 , since SA is the resultant of 
P, and P 2 . 

It is evident that the fourth pressure necessary to produce 
an equilibrium with P x , P 2 , P 3 , being equal and opposite to 
their resultant, is represented in magnitude and direction 
by AE. 

13. Three pressures, P„ P 2 , P 3 , being in equilibrium, it is 
required to determine the third P 3 in terms of the other 
two, and their inclination to one another. 

Let AP X and AP 2 represent the pressures P, and P 2 in 
magnitude and direction, and let the inclination 
I *^^% Pi AP 2 of P, to P 2 be represented by ,0 2 . Com- 
J&C.^ \ plete the parallelogram AP, RP 2 , and draw its 
\..."^ diagonal AR. Then does AR represent the 
* 1 resultant of Pj and P 2 in magnitude and direc- 

tion. But this resultant is in equilibrium with P 3 , since P x 
and P 2 are in equilibrium with P 3 . It acts, therefore, in the 
same straight line with P 3 , but in an opposite direction, and 
is equal to it. Since then AR represents this resultant in 
magnitude and direction, therefore RA represents P 3 in mag- 
nitude and direction. 



Now, AR'^AP, 2 — 2AP, . P X R . cos. AP.R + ^R 2 ; 
also, AP x R=:7r— P x AP 2 =7r— 2 , P^AP,, and AP a , AP 2 , 
AR, represent P„ P 2 , P 3 , in magnitude. 

.-. P/^P^— 2P X P 2 cos. (tt— 2 )+P 2 3 . 
Now cos. (*—A)= —cos. ft,, ;. P^P^ + SPJ 3 , cos. A + P/, 



P 3 = l /P 1 2 + 2P 1 P 2 cos. A + P 2 2 (13). 



14. If three pressures, P„ P 2 , P 3 , be in equilibrium, any two 
of them are to one another inversely as the sines of their 
inclinations to the third. 

Let the inclination of P x to P 3 be represented by ft z , and 
that of P 2 to P 3 by 2 3 . 

Now P 1 AR=tt — P 1 AP 8 =7r-— , 3 , ;. sin. P,AR=sin. ft, ; 
P 1 RA=P,AR=7r— P 9 AP,=7r— A, .-. sin. P/RA^sin. ft,. 



1*'- PARALLEL PRESSURES. 

A1 A IV AI\ Bin. P i:.\ 

Al\~P,R~6in. P,AR J 

• r i _ sin - A (u) 

" r;-.m. a ; ' 

Thar is, P, i> to P, inversely, as tlie sine of the inclina- 
tion of P to P ifl to the sine of the inclination of P, to P 3 . 
Therefore, &e. &c. [q. e. d.] 

Of Parallel Pressures. 

15. Th principle of the equality of moments obtains in 
re% rures in the sam> plane whatever may be 

their inclinations to one another, and therefore if their 
'/motions be infinitely small, or if they be parallel. 

In this case of parallel pressures, the same line AB, which 
^ r T s is drawn from a given point xY, perpendicular 
•V 8 to one of these pressures, is also perpendicular 

to all the rest, so that the perpendiculars are 
' f?l /"here the parts of this line AM,, AM,, &c 
^ ^ * intercepted between the point A and the direc- 
tions of the pressures respectively. The principle is not how- 
ever in this case true only in respect to the intercepted parts 
of this perpendicular line AB, but in respect to the inter- 
cepted parte of any line AC, drawn through the point A 
across the directions of the pressures, since the intercepted 
parte Am,, Am .. Am,, Arc. of this second line are proportional 
to those, AM. .'AM,. & c . of the first. 

Thus taking the case represented in the figure, since by 
tlic principle of the equality of moments we have, 

AM, . P,+AM 4 . P 4 =AM , . P 9 + AH 3 P 3 + AlT 5 P B ; 
dividing both sides by AM,, 

A M^_ AM 4 __ AM 2 AM, 

AM, * Pl + AM, * P «"" AM, * P * + AM, * P ' + P >; 
_ _ . ._ _ AM, Am, AM 2 Am n 

Lutbysmnlartnangles, JJ£= I £ aALT^A^,' **=** 

Am, Am, ? _A^ Am 3 

" A//r ' ' Am, * A ~Am b ' 2_t " Am, ' r *~ i ~ r >- 
Therefore multiplying by Aw,, 

Am,. P,+ Ara 4 . Y t =Am, . P, + Am s . P s + Am 8 . P 5 . 
Therefore, &c [q.e.d.] 



OF PARALLEL PRESSURES. 17 



16. To find the resultant of any number of parallel pressures 

in the same plane. 

It is evident that if a pressure equal and opposite to the 
resultant were added to the system, the whole would be in 
equilibrium. And being in equilibrium it has been shown 
(Art. 8.), that if the pressures were all moved from their 
present points of application, so as to remain parallel to their 
existing directions, and applied to the same point, they are 
such as would be in equilibrium about that point. But 
being thus moved, these parallel pressures would all have 
their directions in the same straight line. Acting therefore all 
in the same straight line, and being in equilibrium, the sum 
of those pressures whose tendency is in one direction along 
that line must equal the sum of those whose tendency is in 
the opposite direction. Eow one of these sums includes the 
resultant R. It is evident then that before R was introduced 
the two sums must have been unequal, and that R equals the 
excess of the greater sum over the less ; and generally that if 
2P represent the sum of any number of parallel pressures, 
those whose tendency is in one direction being taken with 
the positive sign, and those whose tendency is in the opposite 
direction, with the negative sign ; then 

R = HP (15). 

the sign of R indicating whether it act in the direction of 
those pressures which are taken positively, or those which are 
taken negatively. 

Moreover since these pressures, including R, are in equi- 
librium, therefore the sum of the moments about any point, 
of those whose tendency is to communicate motion in one 
direction, must equal the sum of the moments of the rest — 
these moments being measured on any line, as AC ; but one 
tii v5 B of these sums includes the moment of R; these 
two sums must therefore, before the introduc- 
tion of R, have been unequal, and 1 lie moment 
"S[Y*\y\ * of R must be equal to the excess of the greater 
5 sum over the less, so that, representing the 
sum of the moments of the pressures (U nol being included) 
by 2 m 1 P, those whose tendency is to communicate motion 
in one direction, having the positive sign, and the rest the 
negative; and representing by x the distance from A, mea- 
sured along the line AC, at which K intersects that line, we 
have, since xR is the moment of R, xll — 2 m' P, where tho 

2 



l ^ r —-ii\—.l,-i,\— ,;}■„,:- 



18 OP PARALLEL PRESSURES. 

sign of xR indicates the direction in which R tends to turn 
the system about A, but R = 2P, 

•» = ^I (16). 

2P 

Equations (15) and (16) determine completely the magni- 
tude and the direction of the resultant of a system of parallel 
pressures in the same plane. 

17. To determine the resultant of any number of parallel 
pressures not m the same plane. 

Let P, and P, be the points of application of any two of 
these pressures, and let the pressures themselves 
JJS^^ be represented by P, and P Q . Also let their 

^ resultant ^ intersect the line joining the points 

P, and P Q in the point R, ; produce the line 
P 1? P a , to intersect any plane given in position, 
in the point L. Through the points P„ P,, and R,, draw 
P,M„ P 9 M„ and I^IST, perpendicularly to this. plane: these 
lines will be in the same plane with one another and with 
Pj L ; let the intersection of this last mentioned plane with 
the first be LM l5 then will ^M,, P a M 2 , and R,]^ be per- 
pendiculars to LMj ; moreover by the last proposition, 

p 1 tjp i +p 1 d?;=b i ik;; 

" r >LR J +r »' LUr '' 

But by similar triangles, 

LP, P.M, LP a P 2 M 2 




5 



LR, R,1S\ LR^KJN", 

•" '"R^, R^ ' 

Let now the resultant, R 2 , of R x and P, 
intersect the line joining the points R, and 
P 9 in the point R 2 , and similarly let the 
resultant, R 3 , of R 2 and P 4 intersect the 
\ I ine joining the points R 2 and P 4 in the 

x 9 point R 3 , and so on : then by the last equa- 
tion. 



-*1 



OF PARALLEL PRESSUBES. .19 



P, . PM.+P, . PjM, = E, E,N, . 
Similarly, E, . EN 1+ P, . PM, = E, BJX„ 
E, . E,N 2 + p. . P,M, = E 3 EX, 

&c. + &c. = &c. 

&»-, . E^NZ S +P„ . P»M„=E„_ 1 . E^_, JS„_ r 

Adding these equations, and striking out terms common t( 
both sides, 

P,.PM;+PPS+ ... +P„.PS„=E_ I .E^N^ 1 (17) 
Now, _Bi=P,+P« £=B I +P 1 =P 1 +P,+p„ 
E s z=E a + P.=P, + P, + P. + P„ &c. =&c. 

E— ,=P, + P.+P.+ ..... +P»; 

B-i N_ . P,+P,+P.+,fc£+P.=P, . pM,+p . 



P,M,+ +P„ . P„M 



■n 5 



; • K »- 1 ^ 1 - p 1+ p 2 +p 9+ . . . +p n ( 18 ); 

in which expression those of the parallel pressures P , P 
<fec. which tend in one direction, are to be taken positively 
whilst those which tend in the opposite direction are to be 
taken negatively. 

The line E n _ x N"„_i represents the perpendicular distance 
from the given plane of a point through which the resultant 
of all the pressures P 15 P a . . . . P WJ passes. In the same 
manner may be determined the distance of this point from 
any other plane. Let this distance be thus determined in 
respect to three given planes at right angles to one another. 
Its actual position in space will then be known. Thus then 
we shall know a point through which the resultant of all the 
pressures passes, also the direction of that resultant, for it is 
parallel to the common direction of all the pressures, and we 
shall know its amount, for it is equal to the sum of all the 
pressures with their proper signs. Thus then the resultant 
pressure will be completely known. The point K^ is called 
me Centre of Parallel Pressures. 

18. The product of any pressure by its perpendicular dis- 
tance from a plane (or rather the product of the number of 

l unite in the pressure by the number of units in the perpen- 
dicular), is called the moment of the pressure, in respect to 

| that plane. Whence it follows from equation (17) that the 
sum of the moments of any number of parallel pressures in 



20 mi: I BRIBE OF GHAT nv. 

respect to a given plan is equal to the moment of their 
resultant in respect to that plan-. 

19. It is evident, from equation (18), that the distance 
lv, t _i Nn_i of th< of pressure of any number of 

parallel pressures from a given plane, is independent of the 
directions of these parallel pressures, and is dependent 
wholly upon their amounts and the perpendicular distances 
P a M 1 , 1'oM,, &c. of their points of application from the 
given plane. 

So that if the directions of the pressures were changed, 
provided rhat their amounts and points of application 
remained the same, their centre of pressure, determined as 
above, would remain unchanged ; that is, the resultant, 
although it would alter its direction with the directions of 
the component pressures, would, nevertheless, always pass 
through the same point. 

The weights of any number of different bodies or different 
parts of the same body, constitute a system of parallel pres- 
sures ; the direction, therefore, through this system of the 
resultant weight may be determined by the preceding pro- 
position ; their centre of pressure is their centre of gravity. 



The Centre of Gravity. 

20. The resultant of the weights of any number of bodies 
or parts of the same body united into a system of inva- 
riable form passes through the same point in it, into what- 
ever position it may be turned. 

For the effect of turning it into different positions is to 
cause the directions of the weights of its parts to traverse 
the heavy body or system in different directions, at one time 
lengthwise for instance, at another across, at another 
obliquely y and the effect upon the direction of the resultant 
weight through the body, produced by thus turning it into 
different positions, and thereby changing the directions in 
which the weights of its component parts traverse its mass, 
is manifestly the same as would be produced, if without alter- 
ing the position of the body, the direction of gravity could 
be changed so as, for instance, to make it at one time tra- 
verse that body longitudinally, at another obliquely, at a 
third transversely. J hit by Article 19, this last mentioned 
change, altering the common direction of the parallel pres- 



THE CENTRE OF GRAVITY. 21 

sures through the body without altering their amounts or 
their points of application, would not alter the position of 
their centre of pressure in the body / therefore, neither would 
the first mentioned change. "Whence it follows that the 
centre of pressure of the weights of the parts of a heavy 
body, or of a system of invariable form, does not alter its 
position in the body, whatever may be the position into 
which the body is turned; or in other words, that the 
resultant of the weights of its parts passes always through 
the same point in the body or system in whatever position 
it may be placed. 

This point, through which the resultant of the weights of 
the parts of a body, or system of bodies of invariable form, 
passes, in whatever position it is placed ; or, if it be a body 
or system of variable form, through which the resultant 
would pass, in whatever position it were placed, if it became 
rigid or invariable in its form, is called the Centre of 
Gravity. 

21. Since the weights of the parts of a body act in 
parallel directions, and all tend in the same direction, there- 
fore their resultant is equal to their sum. Now, the result- 
ant of the weights of the parts of the body would produce, 
singly, the same effect as it regards the conditions of the 
equilibrium of the body, that the weights of its parts 
actually do collectively, and this weight is equal to the sum 
of the weights of the parts, that is, to the whole weight of 
the body, and in every position it acts vertically downwards 
through the same point in the body, viz. the centre of 
gravity. Thus then it follows, that in every position of the 
body and under every circumstance, the weights of its parts 
produce the same effect in respect to the conditions of its 
equilibrium, as though they were all collected in and- acted 
through that one point of it — its centre of gravity* 

* That the resultant of the weights of all the parts of a rigid body passes 
in all the positions of that body through the same point in it is a property of 
many and most important uses in the mechanism of the universe, as well as in 
the practice of the arts; another proof of it is therefore subjoined, which 
may be more satisfactory to some readers than that given in the text. The 
system being rigid, the distance T\, P 2 , of the points of 
application of any two of the pressures remains the 
same, into whatever position the body may be turned: 
the only difference produced in the circumstance under 

\ which they are applied is an alteration in the inclina- 
tions of these pressures to the line 1\, P 9 : now being 
weights, the directions of these pressures always remain 
parallel to one another, whatever may be their inclina- 
tion; thus then it follows by the principle of the equa- 



? " 11(1 
O r \\ V 



22 THE CENTRE OF GRAVITY. 



22. To determine tlie position of the centre of gravity of 
two weights, P, and reforming part of a rigid system. 

Let it be represented by G. Then since the resultant of 

( P, and P 2 passes through G, we have by equa- 

*' fr "^ P " tion (16), taking P 1 as the point from which the 
moments are measured, 



P 1 +P - .P,G=P JL1 PA 

p pp 

" r ^~ P. + P, » 

whence the position of G is known. 

23. It is required to determine the centre of gravity of three 
weights P„ P 2 , P 3 , not in the same straight line, andfoi v m- 
ing part of a rigid system. 



Find the centre of gravity G„ of P, and P 2 , as in the last 

proposition. Suppose the weights P, and P 2 to 

*rt be collected in G n and find as before the com- 

J& mon centre of gravity G 2 of this weight P, + P„ 

v?^^*>\ so collected in G„ and the third weight P 3 . It 

L is evident that this point G 2 is the centre of 

gravity required. Since G 2 is the centre of 

gravity of r 3 and P, + P 2 collected in G„ we have by the 

last proposition 

G^ 2 . P 1 +P 2 +P 3 ~G^ 3 • P„ 
G 1 P 3 .P 3 



G,G„= 



>-p i+ p a +p 3 - 



lity of moments (Art. 15), that Pi+Pa • PiRi=P 2 . PiP 2 , so that for every 
such inclination of the pressures to Pi P 2 , the line PiPn is of the same length, 
and the point Pm therefore the same point ; therefore, the line P 3 Ri is always 
the same line in the body; and R t which equals P,-j-P 2 , is always the same 
pressure, as also is P 3 , and these pressures always remain parallel, therefore, 
for the same reason as before, R s is always the same point in the body in 

whatever position it may be turned, and so of R 3 , R 4 and R,-,. that 

Is, iii every position of the body, the resultant of the weights of its parts 
passes through the same point Riu.] in it. Since the resultant of the weights 
of the parts of a body always passes through its centre of gravity, it is 
evident, that a single force applied at that point equal and opposite to this 
resultant, that is, equal in amount to the whole weight of the body, and in a 
direction vertically upwards, would in every position of the body sustain it. 
This property of the centre of gravity, viz. thai it is a point in the body where 
a single force would support it is sometimes taken as the definition of it. 



OF A TRIANGLE. 23 

If P 15 P 2 , P 3 , be all equal, then 
Moreover in this case, 

24. To find the centre of gravity of four weights not in thf 
same plane. 

Let P x , P 2 , P 3 , P 4 , represent these weights; find the 

centre of gravity G 2 of the weights P l5 P.,, 

» P 3 , as in the last proposition ; suppose these 

,4 \ three weights to be collected in G 2 , and then 

//| \ find the centre of gravity G 3 of the weight 

/VS.. \ tnus collected in G 2 and P 4 . G 3 will be the 

^i—^-.-.^ss*^' centre of gravity required, and since G 3 is 

the centre of gravity of P 4 acting at the 

point P 4 , and of Pj+Pa+Pg collected at G 2 , 



pp T> 

If all these weights be equal, then by the above equation, 
G 2 G 3 =i G 2 P 4 , 



also, _G 1 G 2 =i_G i P 3 , 

and G^-i P,P 2 . 



25. THE CENTRE OF GRAVITY OF A TRIANGLE. 

Let the sides AB and BC of the triangular lamina ABC 
be bisected in E and D, and the lines CE and 
AD drawn to the opposite angles, then is the 
intersection G of these lines the centre of gravity 
of the triangle : for the triangle may be supposed 
to be made up of exceedingly narrow rectangular 
strips or bands, parallel to BC, each of which will 
be bisected by the line AD; for by similar triangles 
PK : DB:: ATI : AD::EQ : DC, therefore, alternando, 
PR : EQ::DB : DC; but DB=DC; therefore PR=BQ. 

Therefore, each of the elementary bands, or rectangles 
parallel to BC, which compose the triangle ABC, would 
separately balance on the line AD ; therefore, all of theni 




24 -Jin ran 

•!• would balance on the line AD. therefore the 
centre of gravity of the triangle is in AD. 
In the rain- manner it may be shown that t! 
:' the triangle is in the line CE : t3 
rity ie at tne intersection G of tin 
.v I )( ; = l DA : for imagine the triangle to bo without 
ir, and three equal weights to be placed at the ai 
A. i'.. and ('. then it > I that these three weights will 

balance upon AD: for AD being supported, the weight A 
will be supported, Bince it is in that line; mor< B and 

( will be supported Bince they art- equidistant from that 
line. 

Since, then, all three of the weights will balance upon 
AD. their centre of gravity is in AD. In like manner it 
may he- shown that the centre of gravity of all three weights 
is iii GE : therefore it is in &, and coincides with the centre 
of gravity of the triangle. 

Now, suppose the weights B and C to be collected in their 
centre of gravity D, and suppose each weight to be repre- 
sented in amount by A. a weight equal to l'A will then be 
collected in D. and a weight equal to A at A, and the centre 
■avity of these is in G: therefore DAxA = DGx 
A), 

. • . D A = 3 DG, or DG = -J- DA * [q.e.d.] 



26. THE CENTRE OF GRAVITY OF THE PYBAMTD. 

Let ABC be a pyramid, and suppose it to be 

made up of elementary laminae bed, parallel to 
the base BCD. Take'G, the centre of gravity 
of the base BCD, and join AG; then AG will 
pass through the centre of gravity g of the 
lamina bedrf therefore each of the laminae will separately 
balance on the straight line AG : therefore the laminae when 
combined will balance upon this line; therefore the whole 
figure will balance on AG. and the centre of gravity of 
the whole is in AG. In like manner if the centre of gravity 
71 of the face A ! »1) he taken, and CH be joined, then it may 
own that the centre of gravity of the whole is in CH ; 

• Note ((f) Ed. Apn. 

;■ produce the plane ABG to intersect tlio plane ADC in AM, then by 
similar triangles I'M : \iC::dm : m«,but PM = MO; therefore cbn=me. Also 
by similar triangles GM : BM::^m : 6m, but GM = 1 BM ; therefore gm=\ 
hm. Since then dm — \ 4c and gm = i bm, therefore g is the centre of gravitj 
of the triangle bdc. 




OF A TYIiAMID. 25 

therefore the lines AG and CH intersect, and the centre of 
gravity is at their intersection K. 

Now GK is one-fourth of GA ; for suppose equal weights 
to be placed at the angles A, B, 0, and D of the pyramid 
(the pyramid itself being imagined without weight), then 
will these four weights balance upon the line AG, for one 
of them, A, is in that line, and the line passes through the 
centre of gravity G of the other three. 

Since, then, the equal weights A, B, C, and D balance 
upon the line AG, their centre of gravity is in AG ; in the 
same manner it may be shown that the centre of gravity of 
the four weights is in CH, therefore it is in K, and coincides 
with the centre of gravity of the pyramid. 

Now let the number of units in each weight be repre- 
sented by A, and let the three weights B, C, and D be 
supposed to be collected in their centre of gravity G ; the 
four weights will then be reduced to two, viz. 3A at G, and 
A at A, whose common centre of gravity is K, 

.'. GKx3ATA = GAxA, 

/. 4GK = GA or GK = J GA * [q.e.d.] 

27. The centre of gravity of a pyramid with a polygonal hose 
is situated at a vertical height from the base, equal to one 
fourth the whole height of the pyramid. 

For any such pyramid ABCDEF may be supposed to 
be made up of triangular pyramids ABCF, 
ACDF, and ADEF, whose centres of gravity 
G, H, and K, are situated in lines AL, AM. 
and AN, drawn to the centres of gravity L, M, 
and N of their bases ; LG being one-fourth of 
LA, Mil one-fourth of MA, and NK one-fourth 
of NA. The points G, H, and K, are therefore in a plane 
parallel to the base of the pyramid, and whose vertical dis- 
tance from the base equals one-fourth the vertical height of 
the pyramid. 

Since then the centres of gravity G, II, and X of the ele- 
mentary triangular pyramids wlricn compose the whole poly- 
gonal pyramid are in this plane, therefore the centre of gravity 
of the whole is in this plane, i.e. the centre of gravity of the 
whole polygonal pyramid is situated at a vertical height from 
tin- has*', equal to one fourth the vertical height of the whole 

* Note (h) Ed. App. 





2b THE CENTRE OF GRAVITY 

pyramid, or at a vertical depth from the vertex, equal to three 
fourths of the whole. Now the above proportion is true, 
whatever be the number of the Bides of the polygonal base, 
and therefore if they be infinite in number; and therefore it 
is true of the cone, which may be considered a pyramid hav- 
ing a polygonal base, of an infinite number of Bides : and it 
is true whether the cone or pyramid be an oblique or a right 
OOM or pyramid. 

28. If a body be of a prismatic form, and symmetrical 
about a certain plane, then its whole weight may be sup- 
posed to be collected in the surface of that plane, and uni- 
formly distributed through it. For let 
ACBEFD represent such a prismatic 
body, and abc a plane about which it is 
symmetrical : take /n. an element of uni- 
form thickness whose sides are parallel to 
B the sides of the prism, and which is 
terminated by the faces ACB and DFE of the prism ; 
it is evident that this element m will be bisected by the 
plane abt\ and that its centre of gravity will therefore 
lie in that plane, so that its whole weight may be sup- 
posed collected in that plane ; and this being true of 
every other similar element, and all these elements be- 
ing equal, it follows that the whole weight of the body 
may be supposed to be collected in and uniformly dis- 
tributed through that plane. It is in this sense only that we 
can speak with accuracy of the weight and the centre of gra- 
vity of a plane, whereas a plane being a surface only, and 
having no thickness, can have no weight, and therefore no 
centre of gravity. In like manner when we speak of the 
centre of gravity of a curved surface, we mean the centre of 
gravity of a body, the weights of all whose parts may be sup- 
posed to be collected and uniformly distributed throughout 
that curved surface. It is evident that this condition is 
approached to whenever the body being hollow, its material 
is exceedingly thin. Its whole weight may then be conceived 
t<> be collected in a surface equidistant from its two external 
surfaces. In like manner an exceedingly thin uniform curved 
rod may be imagined t«> have its weight collected uniformly 
in a line passing along rhe centre of its thickness, and in this 
we may speal of the centre of gravity of a line, 
although a line having no breadth or thickness can have nc 
weight, and therefore no centre of gravity. 




t> zus sta.'u.t 



OF ANY QUADRILATERAL FIGURE. 27 



29. THE CENTRE OF GRAVITY OF A TRAPEZOID. 

Let AD and BC be the parallel sides of the trapezoid, of 
which AD is the less. Let AD be represented 
b y a, BC by b, and the perpendicular distance 
]STL of the two sides by A. Draw DE parallel 
to AB. Let G-j be the intersection of 
the diagonals of the parallelogram ABED, 
then will G 2 be the centre of gravity of that parallelo- 
gram. Bisect CE in L, join DL, and take DG 2 =f DL, 
then will G 2 be the centre of gravity of the triangle DEC. 
Draw G 1 M 1 and G 2 M 2 perpendiculars to AD ; then since 
AG^-J- AE, therefore G.M^i FE=J A. And since 
DG 2 = f DL, therefore G 2 M 2 = f EX = f A. Suppose the 
whole parallelogram to be collected in its centre of gravity 
G x , and the whole triangle in its centre of gravity G 2 . Let 
G- be the centre of gravity of the whole trapezoid, and draw 
GM perpendicular to AD. Then would the whole be sup- 
ported by a single force equal to the weight of the trapezoid 
acting upwards at G. Therefore (Art. 17), 



MG . ABCD = G 1 M 1 . ABED + G 2 M . CED 
Now, ABCD = i A (a+b), ABED = ha, 

CE D = i h (b-a), G.M, = i h, G 2 M 2 = f A, 

.-. MG . J A (a+J) = iA. Aa+f A . ih{b— a), 

.\ MG (a+b) = ha+% A (b—a) = i A (a + 2b\ 

...MG = iA.^± 2 ^ (19). 

a a+b v ; 



30. THE CENTRE OF GRAVITY OF ANY QUADRILATERAL FIGURE. 

Draw the diagonals AC and BD of any quadrilateral figure 
ABCD, and let them intersect in E, 
and from the greater of the two parts, 
BE and DE, of either diagonal BD set 
off a part BF equal to the less part. 
Bisect the other diagonal AC in H, join 
HF and take HG equal to one third oi 

HF ; then will G be the centre of gravity of the whole 

figure. 

For if not, let g be the centre of gravity, join 1 1 1 > and III) 

and take HG, = i HB and HG, = £ HD, then will G, and 

G 2 be the centres of gravity of the triangles AJBC and ADC 




28 ] in. I i \ : BE OF GRAVITY. 

respectively (Art. 25). Suppose these triangles to be col 
■ I in their centres of gravity G, <>.: i; is evident that 
the centre of gravitj y. of the whole figure, will be in tliu 
straight line joining the points G G,: L this line intersect 
AC in K; then since a pressure equal to the weight of the 
whole figure acting upwards at g, will be in equilibrium with 
the weights of the triangles collected inG, and G.^, we have, 
by the principle of the equality of moments (Art. 15), 



Kg . ABCD = KG 1 . ABC — KG, . AUG. 

Now since HG, = i IIB, and 1IG, = } BD, therefore G X G, 
is parallel to DB, therefore KG, = -J- BE, and KG„ = -J DE. 
Now let the angle AED = BEC = i. Therefore the perpen- 
dicular from B upon AC = BE sin. », and that from D = DE 

sin. », therefore area of triangle ABC = £ AC . BE sin. i, 
and area of triangle ADC — J AC . DE sin. «, therefore area 
of quadrilateral ABCD = i AC . BE sin. i+i AC . DE 
sin. t = i (BE + DE) AC sin. i. Substituting these values in 
the preceding equation, 

Kg. i (BE+DE) AC sin. i = fr BE . J AC . BE sin. i — 
i DE . i AC . DE sin. », 
_r. K^(BE + DE) = i (BF-DF), 

^^=*==-^=*(BE-DE) = i(BE-BF) = iFE. 
BE+DE 

But since HG = i HF, .-.££ = } FE, .\B^ = KG; that 
is, the true centre of gravity g coincides with the point G. 
Therefore, &c. [q.e.d.] 

*31. In the examples hitherto given, the centre of pressure 
of a system of weights, or their centre of gravity, has been 
determined by method- which are indirect as compared with 
the direct and general method indicated in Article IT. That 
method supposes, however, a determination of the sum of the 
moments of the weights of all the various element- of the 
body in respect to three given planes. Nowin a contirmaus 
body these elements are infiniti in number, each being infi- 
nitely small ; this determination supposes, therefore, the sum- 
mation of an infinite number of infinitely small quantities, 
and requires an application of the principles of the intregal 
calculus. 

Let ^M be taken to represent any small element of the 



THE CENTRE OF GRAVITY. 



29 



volume M of a continuous body, and x its perpendicular 
distance from a given plane. Then will x& aM represent 
the moment of the weight of this element about that plane, 
{^ representing the weight of each unit of the volume M. 
Let ^x aM represent the sum of all such moments, taken in 
respect to all the small elements, such as aM, which make 
up the volume of the body. Then if G x represent the dis- 
tance of the centre of gravity of the body from the given 
plane ; since ^x^M. represents the sum of the moments of a 
system of parallel pressures about that plane, i^M the sum of 
those pressures, and G, the distance of their centre of pres- 
sure from the plane (Art. 19), it follows by equation (18) that 

AM ^' AM (20). 



fxM 



M 



Now it is proved in the theory of the integral calculus,* 
that a sum, such as is represented by the above expression 
2#aM, whose terms are infinite in number, and each the pro- 
duct of a finite quantity x, and an infinitely small quantity 
aM, and in which M is, as in this case, a function of x (and 
therefore x a function of M), is equal to the definite integral 



/a?i 
xdM. Therefore, generally, 



<c a 



xdM 
_« 2 



M 



Similarly, 



ydM 



G 2 = ^__ 
M 



zdM 

G — ** 



(21). 



M 



In the two last of which equations y and z are taken to repre- 
sent, respectively, the distances of the element aM of the 



* Poisson, Journal dc l'Ecole Polytechnique, 18me cithior, p. 820, or Art. 2, 
in the Treatise on Definite Integrals in the Encyclopaedia Metropolitans by the 
author of this work. See Appendix, note A. 



30 THE CENTRE OF GRAVITY. 

body from two other planes, as x represents its distance from 
the first plane ; and 5, and G, to represent the distances of 
its centre of gravity from those planes. The distances G„ 

d , G„ of the centre of gravity from three different planes 
being thus known, its actual position in space is fully deter- 
mined. These three planes are usually taken at right angles 
to one another, and are then called rectangular co-ordinate 
planes, and their common intersections rectangular co-ordi- 
nate axes. 

If the centre of gravity of the body be known to lie in a 
certain plane, and one of the co-ordinate planes spoken of 
above, as for instance that from which G 3 is measured, be 
taken to coincide with this plane in which the centre of gra- 
vity is known to lie, then G 3 = 0, and the position of the cen- 
tre of gravity is determined by the two first only of the above 
three equations. This case occurs when the body, whose 
centre of gravity is to be determined, is symmetrical about a 
certain plane, since then its centre of gravity evidently lies 
in its plane of symmetry. If the centre of gravity of the 
body be known to lie in a certain line, and two of the co-or- 
dinate planes, those for instance from which G 2 and G 3 are 
measured, be taken so as to intersect one another in that line, 
then the centre of gravity will be in both those planes ; there- 
fore G a = and G 3 = 0, and its position is determined by the 
first of the preceding equations alone. This case occurs 
when the body is symmetrical about a given line ; its centre 
of gravity is then manifestly in that line. 

*32. The centre of gravity of a curved line which lies 
w t holly in the same plane. 

Taking M to represent the length S of such a line, we 
have, by equations (21), 

Gi= ^S, Q,=f*** ...(22), 

Example. — Let it he required to determine the centre of 
yravity of a circidar arc EF. 

The centre of gravity of such an arc is evidently in the 

y radius CA, which bisects it; since the arc 

f is symmetrical about that radius. Take a 

T" '™-n plane Gy perpendicular to this radius, and 

cUr " He -* passing through the centre, to measure the 

\J moments from. Let x represent the dis- 

tance PM of any point P in this arc from 



OF A CURVED LINE. 61 

tMs plane; also let s represent the arc PA, and S the arc 
E AF, a the radius C A, and C the chord EF. 

/. x = PM = CP cos. CPM = CP cos. ACP == a cos. — 

Ob 

is *s 

.-. fmdS=af cos. £ ds=a?fcos. £tf (j)= 2a' sin.( 2 -} 

— JS ■ — |S 

the integral being taken between the limits -JS and — JS, 
because these are the values of s which correspond to the 
extreme points F and E of the arc. 

Now 2a sin. \ (— ) = chord of EAF = C, :.JxdS = aC, 

.^4 w 

The distance of the centre of gravity of a circular arc from 
the centre of the circle is therefore a fourth proportional to 
the length of the arc, the length of the chord, and the radius 
of the arc. 

*33. The centre of gravity of a curvilinear area 
which lies wholly in the same plane. 

Let BAC represent such an area. If x and y represent 

the perpendicular distances PN and PM of any 

"I p ^^^n point P in the curve AB from planes AC and 

|7[ f AD, perpendicular to the plane of the given area 

Y and to one another, and M represent the area 

JL LA — J PAM, then, considering this area to be made up 

of rectangles parallel to PM, the width of each 

of which is represented by the exceedingly small quantity 

a#, the area AM of each such rectangle will be represented 

by y&x, and its moment about AD by v-xy&x. 

xydx 

Therefore by equation (20), G, = — ?f — = -tt — • • (24). 

A similar expression determines the value of G 2 ; but one 
more convenient for calculation is obtained, if we consider 
the weight of each of the rectangles, whose length is y, to 
be collected in its centre of gravity, whose distance from AC 



32 



THE « : LVTTY. 



is-Jy. The momeni of the weighl of each rectangle about 

AC will then be represented \yfxr\ whence ii follows that 



a 



W* i ft" 



M-M 



M 



(25). 




\mi»lk. — Suppose the curve APB to £t 5 aparalola, whose 
is AC. 

By the equation to the parabola y 1 = -bo*, if a 
be the distance of the focus from the vertex. 
Moreover, the limits between which the integral 
is to be taken are and a\ and and y„ since at 
A , x — 0,y — 0, and at C, x = x„ y = y x , 

•herefure Pxydx = 2 \/ a /*x%dx=-= */ ax 1 f ; also, ~N. = /*ydx 

a? 

«* 4 3 

= 2 */a fx\dx = = 4/flKCjf, therefore G x = = »,. 

o 
Also, Cy'dx = 4a Cxdx— 2ax* =fp andM=- |/ aa?,^ — iL, 



»S 



therefore G„ = -?/.. 

If, then, G be the centre of gravity of the parabolic area 

ACB, then AH = ? AC. HG = ? CB. 

5 ' 8 



* 34. The cextre of gravity of a surface of revolution. 



Any surface of revolution BxVC is evidently symmetrical 
about its axis of revolution AD, its centre of 
gravity is therefore in that axis. Let the mo- 
ments be measured from a plane passing through 
A and perpendicular to the axis AD, and let x 
and y be co-ordinates of any point P in the 
generating curve APB of the surface, and s the 
_ h of the curve AP. Then M being taken to represent 
the area of the surface, and being supposed to be made up 
of bands parallel to PQ, the area aM of each such band is 
represented (see Art. 40.) •'■ by -~y±s, and its moment by 
QrrnwyAs, 

* Church's Diff. Calculus, Art. 91. 




OF A SURFACE. 33 

s, 

%nj xyds 
Q 2nZxyAs _S, (26) 

Example. — To determine the centre of gravity of the sv/r- 
face of any zone or segment of a sphere. 

Let I^ACi represent the surface of a sphere, 
whose centre is D, and whose radius DP is repre- 
sented by #, and the arc AP by s. Then x = DM 

= DP cos. PDM = cos. i, 2/ = PM = DP sin. 

a 

TDM=a sin. -, .*. 2xy = 2& a sin. - cos. - = a* sin . 

a a a a 




S, 8, 

,\ 2n J xyds == 7r# 2 / sin. — <& 



S a S 9 



i s( 2S, 2SJ 

= f 7T^ 3 J COS. 1_ COS. *- t 

( a a ) 

= W j (l + cos. ^)-( 1 + cos. ^i) 1 

= to" icos. 2 - 2 - cos. a §il (27). 

where S, and S 2 are the values of s at the points 
i3 B, and B 2 , where the zone is supposed to ter- 
minate. 

S, 

Also, since = 2ny, :. M = 2tt t yds 

* i 

s* 

= 2ra /*sin. - ds = fra? 1 cos. §» — cos. ?! I , 
J a (a a) 

,, 1 i B. S.) 

G,= 2«)cos.- + cos.-^ 



S, 



34 



THE CENTRE OF GRAVITY 



=i 



=JJ DE.+ DE, I =DE . 



(28), 



if E be the bisection of E^,. 

If S a = 0, or the zone commence from A, then 



G^-aJl + co 



M.. 



= a cos. 



.8, 



• (29). 



*35. The centre of gravity of a soleo of revolutic or. 

Any solid of revolution BAC is evidently symmetrical 
about its axis of revolution AD, its centre of 
gravity is therefore in that line ; and taking a 
plane passing through A and perpendicular to 
that axis as the plane from which the moments 
are measured, we have only to determine the 
distance AG of the centre of gravity, from 
that plane. 

Xow, if x and y represent the co-ordinates of any point P 
in the generating curve, and M the volume of the portion 
PAQ of this solid, then, conceiving it to be made up of 
cylindrical laminae parallel to PQ, the thickness of each of 
which is Xx\ the volume of each is represented by *y*±x, and 
its moment by r^^xf^x. 




-Zxy'^x 



*Jxtfdx 



M 



M 



(30). 



Example. — To determine the centre of gravity of any solid 
segment of a sphere. 

fi Let B,AC, represent any such segment of a 

sphere whose centre is D and its radius a. Let a 
D and y represent the co-ordinates AM and MP of 
any point P, x being measnred from A ; then by 
the equation to the circle y*=2ax—x*, 

• * fx^dx=*Jx {2ax-x') dx=* (&m\'- iO- 
x, o 

#1 x l 

Also, M=*/^<& = « f{toKB^a?) dx=« (axf-fa*). 



OF THE SEGMENT OF AN ARCH. 35 

1^1 /£z*\ (3a 

If the segment become a hemisphere, x 1 =^a, /.G 1 =-fa. 

36. TA^ centre of gravity of the sector of a circle. 

Let CAB represent such a sector ; conceive the arc ADB 
to be a polygon of an infinite number of sides 
JA and lines, to be drawn from all the angles of the 
c<™-^..i.jjj polygon to the centre C of the circle, these will 
*'^.; \) divide the sector into as many triangles. Now 
j/b the centre of gravity of each triangle will be at 
a distance from C equal to f the line drawn from the vertex 
C of that triangle to the bisection of its base, that is equal 
to f the radius of the circle, so that the centres of gravity of 
all the triangles will lie in a circular arc FE, whose centre is 
C and its radius CF equal to fCA, and the weights of the 
triangles may be supposed to be collected in this arc FE, 
and to be uniformly distributed through it, so that the cen- 
tre of gravity G of the whole sector CAB is the centre of 
gravity of the circular arc FE. Therefore by equation (23), 
if S 1 , C 1 , and a\ represent the arc FE, its chord FE, and its 
radius CF, and S, C, a, the similar arc, chord, and radius of 

ADB, then CG = — — ; but since the arcs AB and FE are 

S 1 
similar, and that a 1 = \a, :. C 1 = |C and S 1 = f S. Substi- 
tuting these values in the last equation, we have 

CG = f^i (32). 

37. The centre of gravity of any portion of a circular ring 
or of an arch of equal voussoirs. 

Let B 1 C 1 C 2 B 2 represent any such portion of a circular ring 

whose centre is A. Let a x represent the 

radius, and C, the chord of the arc BjC,, and 

^j^i'3"" Si its length, and let a a , C 2 similarly represent 

J the radius and chord of the arc B a C a , and S, 

P the length of that arc. 

Also let G, represent the centre of gravity of the sector 

ABjC,, G a that of the sector AB a C 2 , and G the centre of 

gravity of the ring. Then 

AG a x sect. ABA + AG x ring B^BA^ AG; xscct.AB.C, 

Now (by equation 32), AG^f-^i, AG a =$ £i2»; 

S, S a 




6b THE PROPERTIES 

also sector AB.C^JS^/,. Bector AP>.,C,= -js </ . 
.-. ring Bfi&A^ sect. An,(\— sect. AB a C 9 =iS 1 a 1 — £S,a„ 

/. AG . (S I a 1 -S,a 9 )=* (O^-C^), 
/. AG = | °> a *- °> a * (33). 



38. TlFE PROPERTIES OF GULDINU8. 

7^ NL represent any plane area, and AB &<? any <&m, m ^ 

c^m« plane, about which the area is made to 

[▲ N revolve, so thai NL is fry ^A/s revolution made to 

# generate a solid of revolution, then is the volume 
of this solid equal to that of a prism whose base 
is NL, and whose height is equal to the length 
of the path which the centre of gravity G of the 
area NL is made to describe. 

For take any rectangular area PRSQ in NL, whose sides 
are respectively parallel and perpendicular to AB, and let 
MT be the mean distance of the points P and Q, or R and 
S, from AB. Now it is evident that in the revolution of 
NL about AB, PQ will describe a superficial ring. 

Suppose this to be represented by QFPK, let M be the 

centre of the ring, and let the arc subtended by 

/"^j* the angle QMF at distance unity from M be repre- 

q ^C sented by 6, then the area FQPK equals the sector 

* FQM-the sector KPM=iU(^x 0— J MP 3 x 6= 

i& (W-MP 2 )=^( — ±-^?) x(MQ-MP)=d(MTxPQ). 

Now the solid ring generated by PRSQ is evidently equal 
to the superficial ring generated by PQ, multiplied by the 
distance PR. This solid ring equals therefore & (MTxPQ 
x PR) or 6 x MT x PRSQ. Now suppose the area PRSQ 
to be exceedingly small, and the whole area NL to be made 
up of such exceedingly small areas, and let them be repre- 
sented by a } , a„ a„ tvc. and their mean distances MT by a?„ 
x„ <r 3 , &c. then the solid annuli generated by these areas 
respectively will (as we have shown), be represented by 
^x x a x , te 9 a 9 , $x 3 a 3 , &c. &c. ; and the sum of these annuli, 



OF GULDINUS. 37 

or the whole solid, will be represented by ^ 1 « 1 + ^# 2 -f- 
^V&3 + &c, or by d(a? I a 1 + aj a # 9 +# 3 fl 3 + &c.).. Now if ^ repre- 
sent the weight of any superficial element of the plane KL, 
8?^^= the moment of the weight of a 1 about the axis AB, 
x^a^— that of the area a a about the same axis AB, and so 
on, therefore the sum (a? 1 a 1 + a? a a 9 H-a? 8 a, + &c.) f/.=the moment 
of the whole area NL about AB ; but if G be the centre of 
gravity of EX, and GI its distance from AB, then the 
moment of KL about AB=GIxSXu; 
JL therefore the whole solid =& . GI . KL; 

^V^\, but & . GI equals the length of the circu- 
gZ3- ) 1 ^" ar P 8 '^ described by G ; therefore the 
"M^Z"'' V '-/ volume of the solid equals 1STL multi- 
gjjjj^ 1 ^ plied by the length of the path de- 
L v scribed by G, i. e. it equals a prism M, 

/ fc r_ - - z\ whose base is NL, and whose height Gli 

i;i ^iz:z^ "^^g ) is the length of the path described by 
m '"" ^T-^y q. . which i s the first property of GUL- 
DESTTJS. 

39. The above proposition is applicable to finding the 
solid contents of the thread of a screw of variable diame- 
ter, or of the material in a spiral staircase : for it is 
evident that the thread of a screw may be supposed to be 
made up of an infinite number of small solids of revolution, 
arranged one above another like the steps of a staircase, all 
of which (contained in one turn of the thread) might be 
made to slide along the axis, so that their surfaces should all 
lie in the same plane ; in which, case they would manifestly 
form one solid of revolution, such as that whose volume has 
been investigated. The principle is moreover applicable to 
determine the volume of any solid (however irregular may 
be its form otherwise), provided only that it may be con- 
ceived to be generated by the motion of a given plane area, 
perpendicular to a given curved line, which always passes 

through the same point in the plane. For it 
(xSj\^>) is evident that whatever point in this curved 

line the plane may at any instant be traver- 
sing, it may at that instant be conceived to be revolving 
about a certain fixed axis, passing through the centre ot 
curvature of the curve at that point; and tlms revolving 
about a fixed axis, it is generating for an instant a solid of 
revolution about that axis, the volume of which elementary 
solid of revolution is equal to the area of the plane mult i- 




38 HIE PROPERTIES 

plied by the length of the path described by its centre of 
gravity; and tins being true of all such elementary solids, 
each being equal to the product of the plane by the corres- 
ponding elementary path of the centre of gravity, it follows 
that the whole volume of the solid is equal to the product 
of the aria by the whole length of the path. 

4-0. If AB represent any cui^ved line made to revolve about 
the axis AD so as to generate t/ie sur- 
face of revolution BAC, and G l> the 
centre of gravity of this curved line, 
D - then is the area of this surface equal 
to the product of the length of the 
curved line AB, by the length of the 
path described by the point G, during 
the revolution of the cui^ve about AD. This is the second 
property of Guldinus. 

Let PQ be any small element of the generating curve, 

and PQFE a zone of the surface generated by this element, 

this zone may be considered as a portion of the surface of a 

cone whose apex is M, where the tangents to the curve at T 

and V, which are the middle points of PQ and FK, meet 

when produced. Let this band PQFK of the cone QMF be 

developed*) and let PQFK represent its develop- 

/0[ ment ; this figure PQFK will evidently he a circu- 

qs^J lar ring, whose centre is M ; since the develop- 

m ment of the whole cone is evidently a circular 

sector MQF whose centre M corresponds to the 

apex of the cone, and its radius MQ to the side MQ of the 

cone. 

Now, as was shown in the last proposition, the area of 
this circular ring when thus developed, and therefore. of the 
conical hand before it was developed, is represented by 

d . MT . PQ, where 6 represents the arc subtended by QMF 
at distance unity. Now the arc whose radius is MT is 
represented by 6 . MT ; but this arc, before it was developed 
from the cone, formed a complete circle whose radius was 
NT, and therefore its circumference 2«rNT; since then the 
circle has not altered its length by its development, we 
have 

* If the cone be supposed covered with a flexible sheet, and a band such 
as PQFK be imagined to be cut upon it, and then unwrapped from the con« 
and laid upon a plane, it is called the development of the band. 



OF GULDENUS. 39 

Substituting this value of dMT in the expression for the area 
of the band we have 

area of zone PQFK=2^ . NT . PQ. 

Let the surface be conceived to be divided into an infinite 
number of such elementary bands, and let the lengths of 
the corresponding elements of the curve AB be represented 
by * 15 s 2 , s 3 , &c. and the corresponding values of NT by y 1? 
y 2 , y 8 , &c. Then will the areas of the corresponding zones 
be represented by SfryA) 2*yA> 2*y 8 s s , &c. and the area of 
the whole surface B AC by 2*y a + 2rfyA + 2^y 3 s 3 -f- .... or 
by 2*(y 1 8 1 +y 9 8 s +y&+ ....)• But smce ^_i s the centre of 
gravity of the curved line AB, therefore AB . GHf* repre- 
sents the moment of the weight of a uniform thread or wire 
of the form of that line about AD, p being the weight of 
each unit in the length of the line : moreover, this moment 
equals the sum of the moments of the weights s^, s 2 p, s 3 ^ y 
&c. of the elements of the line. 

•'.AB . Qfl^=(yA+yA+yA+ • • • > 
.-.AB . Q-H=y 1 g 1 +yA+y r A+ ■ ■ - ■ 

Therefore area of surface BAC=2*AB . GH=AB 
. (2*GH). 

But 2*GH equals the length of the circular path described 
by G in its revolution about AD. Therefore, &c. 

This proposition, like the last, is true not only in respect to 
a surface of revolution, but of any surface generated by a 
plane curve, which traverses perpendicularly another curve 
of any form whatever, and is always intersected by it in the 
same point. It is evident, indeed, that the same demonstra- 
tion applies to both propositions. It must, however, be ob- 
served, that neither proposition applies unless the motion of 
the generating plane or curve be such, that no two of its con- 
secutive positions intersect or cross one another. 

41. The volume of any truncated prismatic or cylindrical 
body ABCD, of which one extremity CD is perpendicular 
to the sides of the prism, and the other Al> inclined to 
them, is equal to that of an upright prism ABEF, having 
for its base the plane AB, and for its height the perpen- 
dicular height GN of the centre of gravity G of the plane 
DC, above the plane of AB. 

For let < represent the inclination of the plain 4 DC to A B ; 




40 Tin: na of gull-: 

take ///. any small element of the plane 

CD, and let //</• be a prism wh 

and wl - are parallel to AD and 

B( : of elementary prisma Bimilarto which 
■ — -* V®" the whole solid ABCD maybe 
made np. Now the volume of this prism, whose 
and it- height mr, equal? mrxm = E 
— sec » x (mr . sin. mm) m = Bee » x ma > 
Therefore the whole Bolid equals the Bum of all such pro- 
//>. each snch product being multiplied by the 
ant quantity sec. «. or it is equal t<:» the Bum just spoken 
of. that sum being divided by cos. «. Let this Bum be repre- 
sented by Z/nnx?/u therefore the volume of the solid is re- 
presented by — ■ — ■. Now suppose CD to represent a 
eos. ' 

thin lamina of uniform thickness, the weight of each square 
unit of which is m-. then will the weight of the element m be 
represented by . and its moment about the plane AEN 

by u x ran x m. and \^ma x a\ will represent the sum of the 
moments of all the elements of the lamina similar to m about 
that plane. Now by Ait. 15. this sum equals the moment of 
the whole weight of the lamina p x CD supposed to be col- 
lected in G. about that plane. Therefore 
fj. x CD x NG=/a2»ot x ?a, 
.-. CD ■ NG = -aia x a\. 
Substituting this value of ^ma xa. we have 

volume of solid = sec. » x CD x NG. 
But the plane CD is the projection of AB, therefore CD 
= AB cos. >. .*. CD xsec. i = AB ; 

• vol. of solid ABCD = AB xXG = vol. of prism ABEF. 
Therefore, &c. 

[q. e. d.] 



MOTION. 41 



FA.R T II. 
DYNAMICS. 

42. Motion ib change of place. 

The science of Dynamics is that which treats of the laws 
which govern the motions of material bodies, and of their 
relation to the forces whence those motions result. 

The spaces described by a moving body are the distances 
between the positions which it occupies at different succes- 
sive periods of time. 

Uniform motion is that in which equal spaces are de- 
scribed in equal successive intervals of time. 

The velocity of uniform motion is the space which a 
body moving uniformly describes in each second of time. 
Thus if a body move uniformly with a velocity represented 
by Y, and during a time represented in seconds by T, then 
the space S described by it in those T seconds is represented 

by TY, or S=TY. "Whence it follows that Y = Lmd T=^- ; 

so that if a body move uniformly, the space described by it 
is equal to the velocity multiplied by the time in seconds, 
the velocity is equal to the space divided by the time, and 
the time is equal to the space divided by the velocity. 

43. It is a law of motion, established from constant obser- 
vation upon the motions of the planets, and by experiment 
upon the motions of the bodies around us, that when once 
communicated to a body, it remains in that body, unaffected 
by the lapse of time, carrying it forward for ever with the 
same velocity and in the same direction in which it first be- 
gan to move, unless some force act afterwards in a contrary 
direction to destroy it* 

* This is the first law of motion. For numerous illustrations of this fun- 
damental law of motion, the reader is referred to the author's work, entitled 
Illustrations oy Mechanics, Art. 193. 



42 VELOCITY. 

The velocity, at any instant, of a body moving with a 
vabiable motion, is the Bpace which it Mould describe in 

one second of time if its motion wire from that instant to 
become uniform. 

An a« < i i.i.KAii.\(. force is that which acting continually 
upon a body in the direction of its motion, produces in it a 
continually increasing velocity of motion. 

A retarding force is that which acting upon a body in 
a direction opposite to that of its motion produces in it a 
continually diminishing velocity. 

An impulsive force is that which having communicated 
motion to a body, ceases to act upon it after an exceedingly 
small time from the commencement of the motion. 



44. A uniformly accelerating or retarding force is that 
which produces equal increments or decrements of velocity 
in equal successive intervals of time. If f represent the 
additional velocity communicated to a body by a uniformly 
accelerating force in each successive second of time, and T 
the number of seconds during which it moves, then since by 
the first law of motion it retains all these increments of velo- 
city (if its motion be unopposed), it follows that after T 
seconds, an additional velocity represented by yT, will have 
been communicated to it ; and if at the commencement of 
this T seconds its velocity in the same direction was Y, then 
this initial velocity having been retained (by the first law of 
motion), its whole velocity will have become Y+fT. 

If, on the contrary, f represents the velocity continually 
taken away from a body in each successive second of time, 
by a uniformly retarding force, and Y the velocity with 
which it began to move in a direction opposite to that in 
which this retarding force acts, then will its remaining velo- 
city after T seconds be represented by Y— yT; so that gene- 
rally the velocity Y of a body acted upon by a uniformly 
accelerating or retarding force is represented, after T seconds, 
by the formula 

v=V±/T (34). 

The force of gravity is, in respect to the descent of bodies 
near the eartlrs surface, a constantly accelerating force, 
increasing the velocity of their descent by 32£ feet in each 
successive second, and if they be projected upwards it is a 
constantly retarding force, diminishing their velocity by that 
quantity in each second. The symbol g is commonly used to 




VELOCITY. 43 

represent this number '32 } ; so that in respect to gravity the 
above formula becomes v==Y±gT, the sign ± being taken 
according as the body is projected downwards or upwards. 

A variable accelerating force is that which communicates 
unequal increments of velocity in equal successive intervals 
of time ; and a variable retarding force that which takes 
away unequal decrements of velocity.* 

45. To DETERMINE THE RELATION BETWEEN THE VELOCITY AND 
THE SPACE, AND THE SPACE AND TIME OF A BODY'S MOTION. 

Let AMj, M^M^ M 2 M S , &c. represent the exceeding small 
successive periods of a body's motion, and 
AP the velocity with which it began to 
move, M 1 P 1 the velocity at the expiration 
of the first interval of time, M 2 P 2 that at 
the expiration of the second, M 3 P 3 of the 
third interval of time, and so on ; and 
instead of the body varying the velocity of its motion con- 
tinually throughout the period AM l5 suppose it to move 
through that interval with a velocity which is a mean 
between the velocity AP at A, and that ^M. 1 P 1 at M l5 or with 
a velocity equal to ^(AP+l^P,). 

Since on this supposition it moves with a uniform motion, 
the space it describes during the period AM, equals the 
product of that velocity by that period of time, or it equals 
^■(AP + MjP^AMj. Now this product represents the area 
of the trapezoid AM,P X P. The space described then in the 
interval AM n on the supposition that the body moves during 
that interval with a velocity wdiich is the mean between 
those actually acquired at the commencement and termi- 
nation of the interval, is represented by the trapezoidal 
areaAM.P.P. 

Similarly the areas PjM^ P 2 M 3 , &c. represent the spaces 
the body is made to describe in the successive intervals 
M,M„ M 2 M 3 , &c. ; and therefore the whole polygonal area 
APCB represents the whole space the body is made to 
describe in the whole time AB, on the supposition that it 
moves in each successively exceeding small interval of time 
with the mean* velocity of that interval. Now the less the 
intervals are, the more nearly does this mean velocity of each 
interval approach the actual velocity of that interval ; and 
if they be infinitely small, and therefore infinitely great in 

* Note (i) Ed. App. 



44 MOTION UNIFORMLY 

number, then the mean velocity coincides with the actual 
velocity of each interval, and in this case the polygonal area 
passes into the curvilinear area APCB. 

Generally, therefore, it' we represent by the i of * 

curve the times through which a body has moved, and by 
the corresponding ordinate* of that curve the velocities which 
it has acquired after those times, then the a/rea of that curve 
will represent the space through which the body has moved; 
or in other words, if a curve IV be taken such that the num- 
ber of equal parts in any one of its abscissae AM, being taken 
to represent the number of seconds during which a body has 
moved, the number of those equal parts in the corresponding 
ordinate M f P, will represent the number of feet in the velo- 
city then acquired ; then the space which the body has 
described will be represented by the number of these equal 
parts squared which are contained in the area of that curve. 



46. To determent: the space described in a given time by 

A BODY WHICH IS PROJECTED WITH A GIVEN VELOCITY, AND 
WHOSE MOTION IS UNIFORMLY ACCELERATED, OR UNIFORMLY 
RETARDED. 

Take any straight line AB to represent, the whole time T, 
jn in seconds, of the body's motion, and draw AD 
perpendicular to it, representing on the same 
scale its velocity at the commencement of its 
motion. Draw DE parallel to AB, and accord- 
ing as the motion is accelerated or retarded 
draw DC or DF inclined to DE, at an angle whose tangent 
equals/', the constant increment or decrement of the body's 
velocity. Then if an y abscissa AM be taken to represent a 
number of seconds t during which the body has moved, the 
corresponding ordinate MP or MQ will represent the velocity 
then acquired by it, according as its motion is accelerated or 
retarded. For PR = RQ = DE tan. PDE= AM tan. PDE ; 
but AM = £, and tan. PDE=f: therefore PR = KQ=/tf. 
Also RM=AD=V, therefore MP= EM + PB=Y+//, and 
MQ=RM— RQ=V— /*; therefore by equation (34)/MP or 
MQ represents the velocity after the time AM according as 
the motion is accelerated or retarded. The same being true 
of every other time, it follows, by the last proposition, that 
the whole space described in the time T or AB is represented 
by the area ABCD if the motion be accelerated, and by the 
area ABFD if it be retarded 



ACCELERATED OR RETARDED. 45 

Now area ABCD=iAB(AD+BC), but AB=T, AD=Y, 
BC-Y+/T, 

.-. area, ABCD=iT(Y+Y+fT)=YT+ifY. 

Also area ABFD=JAB (AD + BF), where AB and AD 
have the same values as before, and BF=Y— fT> 

:. area ABFD=4T(Y + Y-/T)=YT-i/T\ 

Therefore generally, if S represent the space described after 
T seconds, 

S=YT±ifT* (35); 

in which formula the sign ± is to be taken according as the 
motion is accelerated or retarded. 



47. To DETERMINE A RELATION BETWEEN THE SPACE DESCRIBED 
AND THE VELOCITY ACQUIRED BY A BODY WHICH IS PROJECTED 
WITH A GIVEN VELOCITY, AND WHOSE MOTION IS UNIFORMLY 
ACCELERATED OR RETARDED. 

Let v be the velocity acquired after T seconds, then by 
equation (34), v=Y±f% :. T== ±^—Q- 



= £AB( 



c Now area ABCD = £AB (AD + BC), where 
E AB=T= ( — 9^> AD = Y, BC=<y, 

;. area ABCD = ^~7^ (Y + v) = i^~ V * ) » 

area ABFD = JAB (AD + BF), where AB= T = -^^> 
AD=Y, BF=^. 

... areaABFD^-j^-^^ + ^^-X^. 

Therefore generally, if S represent the space through which 

the velocity v is acquired, then S = ±£- — -j. — -> 

,-.«P— V , = ±2/S (36); 

in which formula the ± sign is to be taken according as 
the motion is accelerated or retarded. 

If the body's motion be retarded, its velocity v will event a- 
ally be destroyed. Let S x be the space which will have been 



46 THE UNIT OF WORK. 

mifihee, thai by the last eqiatijn 
0-V=-2/S.. 

••• V=8/fi 

where A' > the -■ with which the 

in ad id S, the wl. 

which by thk ty it can 

- 

If * - I it fall from 

or Ik. jeetion. then — =+2/^ 

■■• '=»/S 38> 

Let S. be the space through which it must in this case 
move to acquire a velocity V equal to that with which it 
eted in the last case, the: '== _~ S, Whence 

it follows thai v =>.. -" that the whole space 8, through 
which a body will move when pi with a given 

city A', and uniformly - / by any force, is equal to the 
space S,. through which it must move I that velo- 

city when uniformly a taibythe e I roe. 

In the case of - moving freely, and acted upon by 

gravil - _ ■ - and is represented by g; and the 

space S.. through which any given velocity V is acquired, is 
then said to be that d\u: to that velocity. 



WORK 

4-. TTork is the union of a continued pressure with a 
continued motion. And a mechanical agent is thus said to 
work when a pressure is continually overcome, and a point 
< to which that pressure is applied | continually moved by it. 
Neither pressure nor motion alone is sufficient to constitute 
work : so that a man who merely supports a load upon hi9 
shoulders, without moving it. no m fe», in the sense in 

which that term is here used, than does a column which sus- 
tains a heavy weight upon its summit ; and a stone, as it falls 
freely ,\ no more works than do the planets as they 

—red through space.* 

49. The mra of work. — The unit of work used in this 
country, in terms oi which to estimate every other amount 

* RMe ■ j ) Ed. App. 



VARIABLE WORK. 47 

of work, is the work necessary to overcome a pressure of one 
pound through a distance of one foot, in a direction opposite 
to that in which a pressure acts. Thus, for instance, if a 
pound weight be raised through a vertical height of one foot, 
one unit of work is done ; for a pressure of one pound is 
overcome through a distance of one foot, in a direction oppo- 
site to that in which the pressure acts. 

50. The number of units of work necessary to overcome a 
pressure of M pounds through a distance of N feet, is 
equal to the product MN. 

For since, to overcome a pressure of one pound through 
one foot requires one unit of work, it is evident that to over- 
come a pressure of M pounds through tht> same distance of 
one foot, will require M units. Since, then, M units of work 
are required to overcome this pressure through one foot, it 
it evident that K times as many units (i. e. NM) are required 
to overcome it through X feet. Thus, if we take U to repre- 
sent the number of units of work done in overcoming a con- 
stant pressure of M pounds through N feet, we have 

U=M^ (39).* 

51. To ESTIMATE THE WORK DONE TINDER A VARIABLE 
PRESSURE. 

Let PC be a curved line and AB its axis, such that any 
a^- *^ one of its abscissas AM 3 , containing as many 

yfj | c equal parts as there are units in the space 

tA \ through which any portion of the work has 

LLyj Ib been done, the corresponding ordinate M 9 P, 

may contain as many of those equal parts, 
as there are in the pressure under which it is then being 
done. Divide AB into exceedingly small equal parts, AM,, 
M,M 2 , &c, and draw the ordinates M a P„ M 2 P 2 , &c. ; then if 
we conceive the work done through the space AM, (which 
is in reality done under pressures varying from AP to M,P,), 
to be done uniformly under a pressure, which is the arith- 
metic mean between AP and M,P„ it is evident that the 
number of units in the work done through that small space 
will equal the number of square units in the trapezoid 
APP,M, (see Art. 45.), and similarly with the other trape- 

* Note (k) Ed. App. 



48 -OLCTION 

goids; bo that the number of units in the whole work done 
tlirough the space Al> will equal the number of square units 
in the whole polygonal area AIT PI' . & 

But since AM,. -MAI • &a, are exceedingly small, this 
polygonal area passes into the curvilinear ar< a APCB; the 
whole work done is therefore represented by the number of 
square equal parts in this area. 

Now, generally, the area of any curve is represented by 

the integral fydx, where y represents the ordinate, and x 

the corresponding abscissa. But in this case the variable 
'ire P is represented by the ordinate, and the space S 
described under this variable pressure by the abscissa. If 
therefore U represent the work done between the values S, 
and S a of S, we have. 



r 



=/~EV7S (40). 



2fean pressure is that under whicli the same work would 
be done over the same space, provided that pressure, instead 
of varying throughout that space, remained 
p the same : thus, the mean pressure in re- 
F spect to an amount of work represented by 

I the curvilinear area AEFC, is that under 

c which an amount of work would be done 
represented by the rectilineal area ABDC, the area ABDC 
being equal to the curvilinear area AEFC ; the mean pres- 
sure in this case is represented by AB. Thus, to determine 
the mean pressure in any case of variable pressure, we have 
only to find a curvilinear area representing the work done 
under that variable pressure, and then to describe a rectan- 
gular parallelogram on the same base AC, which shall have 
an area equal to the curvilinear area. 

If S represent the space described under a variable pres- 
sure, U the work done, and p the mean pressure, then 

^>S =Uj therefore p ="«•* 

52. To estimate the work of a pressure* who* ion is not 

that hi which its point of appli & mad': to move. 

Hitherto the work of a force has been estimated only od 

* Note (/) Ed. App. 




OF WOBK. 49 

the supposition that the point of applica- 
tion of that force is moved in the direction 
in which the force operates, or in the oppo- 
site direction. Let PQ be the direction of 
a pressure, whose point of application Q 
is made to move in the direction of the 
straight line AB. Suppose the pressure P to remain con- 
stant, and its direction to continue parallel to itself. It is 
required to estimate the work done, whilst the point of 
application has been moved from A to Q. 

Resolve P into E and S, of which R is parallel and S per- 
pendicular to AB. Then since no motion takes place in the 
direction of SQ, the pressure S does no work, and the whole 
work is done by R ; therefore the work = R . AQ. 

]S"ow R= P . cos. PQR, therefore the work =P . AQ cos. 
PQR. Prom the point A draw AM perpendicular to PQ, 
then AQ cos. PQR=QM ; therefore work=P . QM. There- 
fore the work of any pressure as above, not acting in the 
direction of the motion of the point of application of that 
pressure, is the same as it would have been if the point of 
application had been made to move in the direction of the 
pressure, provided that the space through which it was so 
moved had been the projection of the space through which 

it actually moves. The product P . QA1 may be called the 
work of P resolved in the direction of P. 

The above proposition which has been proved, whatever 
may be the distance through which the point of application 
is moved, in that particular case only in which the pressure 
remains the same in amount and always parallel to itself, is 
evidently true for exceedingly small spaces of motion, even 
if the pressure be variable both in amount and direction ; 
since for such exceedingly small variations in the positions 
of the points of application, the variations of the pressures 
themselves, both in amount and direction, arising from these 
variations of position, must be exceedingly small, and there- 
fore the resulting variations in the work exceedingly small 
as compared with the whole work.* 

* Note (m) Ed. App. 




THE WORK OF 

' . P„ P . 

sa? to thsm- 

sel through 

' line AB, then * d to the 

su the 

■ a negat 
- 
totmj 

Let a., a., a v c \:c. represent the inclinations of the pres- 
- -s P.. P.. &c to the line AB. then will 
the resolved parts of these pressures in the 
direction of that line be P. cos. *.. P. 
«.. P 3 c - . and they will be equiva- 

lent * _ m ure in the direction 

of that line represented by P, cos. a- x + P f 
- - _ - I ' - fee. in which sum all 
a terms are to be taken negatively which involve pres- 
sures whose direction is from B to war :*e the single 
genre from A towards B is manifestlv equal to the diner- 
ence between the sun. res which act 
in that direction, and those in the opposite direction). There- 
*he whole work is equal to P. COS. *. — P. cos. ^ — P 5 

- ^_ AB = P. . A B gob. *. - P : . SB « 

+P.AB dob. ■ - . . . =P : . BM.+P, . B.M- P 3 . BM : - 

: in which expression the successive terms are the 

works of the different pressures resolved in the several 
directions of those pressures, each being taker- 
negative' ling as the direction of the corresponding 
ire is towards the direction of the motion or opposite 
to it. 

Thus if U represent the whole work and U x and U, the 
sums of those done in opposite directions, then 

r=u -u,* .41). 



54:. If any ranrJ nt be in equi- 
librium, and tl- t criolt 
work done by these pressure* qftht tm 
22 equal the ichoU work done in tl ■ -ection. 

- if the pressures P .. P.. P.. &e. Ar be in equi- 
librium, then the sums of their - res in opposite 

* Note («) Ed. App. 



CENTRAL FOECES. 51 

directions along AB will be equal (Art. 10) ; therefore the 
whole work TJ along AB, which by the last proposition is 
equal to the work of a pressure represented by the difference 
of these sums, will equal nothing, therefore O^Uj— U 2 , 
therefore U 1 =TJ a , that is, the whole work done in one direc- 
tion along AB, by the pressures P 1? P 2 , &c. is equal to the 
whole work done in the opposite direction. 

55. If a body he acted upon by a force whose direction is 
always towards a certain point S, called a centre of force, 
and be made to describe any given curve PA in a direction 
opposed to the action of that force, and Sp be measured on 
SA equal to SP, then will the work done in moving the 
body through the curve PA be equal to that which would 
be necessary to move it in a straight line from p to A. 

For suppose the curve PA to be a portion of a polygon of 
an infinite number of sides, PP l5 P a P 2 , &c. 
Through the points P, P l5 P 2 , &c. describe circu- 
lar arcs with the radii SP, SP l5 SP 2 , &c. and let 
them intersect S A in p, p^ p 2 , &c. Then since 
PP X is exceedingly small, the force may be consi- 
dered to act throughout this space always in a 
direction parallel to SY 1 ; therefore the work done 
through PP X is equal to the work which must be 
done to move the body through the distance raP, (Art. 52.), 
since mP, is the projection of PP, upon the direction SP X of 
the force. But mP 1 =p2? 1 ; therefore the work done through 
PP X is equal to that which would be required to move the 
body along the line S A through the distance pp 1 ; and simi- 
larly the work done through P 2 P 2 is equal to that which 
must be done to move the body through p x p„ so that 
the work through PP 2 is equal to that through pp„ and so 
of all other points in the curve. Therefore the work through 
PA is equal to that through pA* Therefore, &c. [q.e.d.] 



* Of course it is in this proposition supposed that the force, if it be not 
constant, is dependant for its amount only on the distance of the point at 
which it acts from the centre of force S ; so that the distances of p and F 
from S being the same, the force at p is equal to that at P ; similarly the 
force at j»i is equal to that at Pi, the force at j? 2 equal to that at P 3 , &c. 




■ ■-! THE WOSX OF 

li/ngly <j 
with AP, then aU tht lines drmtmfrom S to A; 

d parallel. This is the case with the force of era 
at the surface of the earth, which tends towards a point, the 
earth's centre, situated at an exceedingly great distant 
c<>mpare<l with any of the distances through which the work 
of mechanical agi imated. 

Thus then it follows that the work nee- move a 

heavy body up any curve PA, or inclined plane, is the same 
•uld be necessary to raise it in a vertical line j?A to the 
same height. 

The dimensions of the body are here supposed to be ex- 
ceeding small. If it be of considerable dimensions, then 
whatever be the height through which its centre of gravity 
is raised along the curve, the work expended is the same 
(Art 60.) as though the centre of gravity were raised verti- 
cally to that height.* 



57. In the preceding propositions the work has been esti- 
mated on the supposition that the body is made to mo* 

increase its distance from the centre S, or in a direction 
opposed to that of the force impelling it towards S. It is 
evident, nevertheless that the work would have been precisely 
the same, if instead of the body moving from P to A it had 
moved from A to P, provided only that in this last case 
there were applied to it at every point such a force as would 
prevent its motion from being accelerated by the force con- 
tinually impelling it towards S : for it is evident that to pre- 
vent this acceleration, there must continually be applied t » 
the body a force in a direction from S equal to that by which 
it is attracted towards it ; and the work of such a force is 
manifestly the same, provided the path be the same, whether 
the body move in one direction or the other along that path, 
g in the two cases the work of the same force over the 
same space, but in opposite directions. 

* The onhi force acting upon the body is in this proposition supposed to be 
that acting towards S. Xo account is taken of friction or any other force* 
which oppose themselves to its motion. 



PARALLEL FORCES. 53 



58. If there he any number of parallel pressures, P„ P 2 , P 8 , 
&c. whose points of application are transferred, each 
through any given distance from one position to another, 
then is the work which would be necessary to transfer their 
resultant through a space equal to that by which their 
centre of pressure is displaced in this change of position, 
equal to the difference between the aggregate work of those 
pressures whose points of application have been moved in 
the directions in which the pressures applied to them act, 
and those whose points of application have been moved in 
the opposite directions to their pressures. 

For (Art. IT.), if y y , y 2 , y s , &c. represent the distances of 
the points of application of these pressures from any given 
plane in their first position, and h the distance of their centre 
of pressure from that plane, and if Y 1? Y 2 , Y 3 &c. and H re- 
present the corresponding distances in the second position, 
and if P 1} P 2 , P 3 , &c. be taken positively or negatively ac- 
cording as their directions are from or towards the given 
plane, h {P 1 + P 2 + P 3 + . . . } =P# 1 +P 2 y 2 +P 9 y 3 • • • • 

andH{P 1 + P i + P i + }=P 1 Y 1 + P 2 Y 2 + P 3 Y 3 + .... 

., QEL-h) {P 1+ P 2 +P 3 + . . . } =P, (Y-2/0+P, (T.-yJ 
+P 3 (Y 3 -y 3 )+. (42); 

in the second member of which equation the several terms 
are evidently positive or negative, according as the pressure 
P corresponding to each, and the difference Y— y of its dis- 
tances from the plane in its two positions, have the same or 
contrary signs. Now by supposition P is positive or negative 
according as it acts from or towards the plane ; also Y—y is 
evidently positive or negative according as the point of appli- 
cation of P is moved from or towards the plane ; each term 
is therefore positive or negative, according as the correspond- 
ing point of application is transferred in a direction towards 
that in which its applied pressure acts, or in the opposite 
direction. 

Now the plane from which the distance's of the points of 
application are measured may be any plane whatever. Let 
it be a plane perpendicular to the directions of the pressures. 




;.4 Tin: work of 

Let Axy represent this plane, and let P 
P' represent the two positions of the point 
of application of the pressure P (the path 
described by it between these two positions 
having been any whatever). Let MP and 
MP' represent the perpendicular dis- 
tances of the points P and P' from the 
plane, and draw F?n from P perpendicular 
to MP'. ThenP (Y-y)=P(MT'-MP)=P . mF] but by 
Art. 55., P . rnV equals the work of P as its point of applica- 
tion is transferred from P to P'. Thus each term of the second 
member of equation (42) represents the work of the corre- 
sponding pressure, so that if 2^,, represent the aggregate 
work of those pressures whose points of application are trans- 
ferred towards the directions in which the pressures act, and 
i '/ the work of those whose points of application are moved 
opposite to the directions in which they severally act, then 
the second member of the equation is represented by Zu x — 
2^ a . Moreover the first member of the equation is evidently 
the work necessary to transfer the resultant pressure P,-t- 
P, + P, &c. through the distance H— /<, which is that by 
which the centre of pressure is removed from or towards the 
given plane, so that if U represent the quantity of work 
necessary to make this transfer of the centre of pressure, 
U=St* 4 — 2t#, (43). 

59. If the sum of those parallel pressures whose tendency 
is in one direction equal the sum of those whose tendency 

is in the opposite direction, then P x + P a + P, + =0. 

In this case, therefore, U=0, therefore 2^— iu^=0, there- 
fore ^u,= Zu n _ ; so that when in any system of par olid pres- 
sures the sum of those whose tendency is in one direction 
equals the sum of those whose tendency is in the opposite direc- 
tion* then the aggregate work of those whose points of appli- 
cation an moved in the directions of the pressure severally 
applied to them is equal to the aggregate work of those whose 
points of application are moved in the opposite direction*. 

This case manifestly obtains when the parallel pressures 
are in equilibrium, the sum of those whose tendency is in 
one direction then equalling the sum of those whose tendency 
is in the opposite direction, since otherwise, when applied to 
a point, these pressures could not be in equilibrium about 
that point (Art. SX 



PARALLEL FORCES. 55 

60. The preceding proposition is manifestly trne in respect 
to a system of weights, these being pressures whose directions 
are always parallel, wherever their points of application may 
be moved. 2sbw the centre of pressure of a system of 
weights is its centre of gravity (Art. 19). Thus then it fol- 
lows, that if the weights composing such a system be sepa- 
rately moved in any directions whatever, and through any 
distances whatever, then the difference between the aggre- 
gate work done upwards in making this change of relative 
position and that done downwards is equal to the work 
necessary to raise the sum of all the weights through a height 
equal to that through which their centre of gravity is raised 
or depressed.* Moreover that if such a system of weight- 
be supported in equilibrium by the resistance of any fixed 
point or points, and be put in motion, ttrn (since the work 
of the resistance of each such point is nothing) the aggregate 

* This proposition has numerous applications. If, for instance, it be required 
to determine the aggregate expenditure of work in raising the different ele- 
ments of a structure, its stone, cement, arc, to the different positions they 
occupy in it, we make this calculation by determining the work requisite to 
raise the whole weight of material at once to the height of the centre of gra- 
vity of the structure. If these materials have been carried up by labourers, and 
we are desirous to include the whole of their labour in the calculation, we 
ascertain the probable amount of each load, and conceive the weight of a la- 
bourer to be added to each load, and then aU these at once to be raised to the 
height of the centre of gravity. 

Again, if it be required to determine the expenditure of work made in rais- 
ing the material excavated from a well, or in pumping the water out of it, we 
know that (neglecting the effect of friction, and the weight and rigidity of the 
cord) this expenditure of work is the same as though the whole material had 
been raised at one lift from the centre of gravity of the shaft to the surfac?. 
Let us take another application of this principle which offers so many practic il 
results. The material of a railway excavation of considerable length is to be 
removed so as to form an embankment across a valley at some distance, and ir 
is required to determine the expenditure of work made in this transfer of th 
material. Here each load of material is made to traverse a different distune 
a resistance from the friction, <£c, of the road being continually opposed to i - 
motion. These resistances on the different loads constitute a system of para- 
lel pressures, each of whose points of application is separately transferred fro i 
one given point to another given point, the directions of transfer being als i 
parallel. Xow by the preceding proposition, the expenditure of work in a'l 
these separate transfers is the same as it would have been had a pressure equ il 
to the sum of all these pressures been at once transferred from the centre o 
resistance of the excavation to the centre of resistance of the embankment 
Xow the resistances of the parts of the mass moved are the frictions of its el' 
ments upon the road, and these frictions are proportional to the weights of th" 
(dements; their centre of resistance coincides therefore with the centre of gra- 
vity of the mass, and it follows that the expenditure of work i< the same al- 
though all the material had been moved at onee from the centre of gravity of 
the excavation to that of the embankment. To allow for the weight of the 
carriages, as many times the weight of a carriage mnst be added to the weight 
of the material as there are journeys made. 



.V; STABILITY OF THE CENTRE, 

work of those weights which are made to descend, is equal 
to that of those which are made to ascend. 

61. If a plant h tafa n perp ndicularto the directions of any 
numbt r of parallel pressures and ///< n be two difft /■> nt po- 
sitions <>/ ih> points of amplication of c\ rtain ofthest j>r>s- 
sures in which they are cti different distances from the 
planes whilst thi points of application qftht rest of then 
pressures remain atthesame distance from thi ft plane, 
and [t in both positions the system h in equilibrium^ then 
the centre of pressure of the first mi ntioru dpr< ssun % will 
be at the sam* distance from the plane in both positions. 

For since in both positions the system is in equilibrium, 
therefore in both positions P,+P 9 + P 3 4- . . . =0, 
.•.(Y-y 1 )P 1 + (Y-yJP 2 + (Y3-y 3 )P3 + ...+P„(Y-y n )=0. 
Now let V be any one of the pressures whose points ot appli- 
cation is at the same distance from the given plane in both 
positions, 

•'• Y n=//„> and Y n- 7 Jn = °> 
.•.CT 1 -y 1 )P 1 +(Y-^)P,+. • . + (Y _- I/n _ 1 )V n _ =0, 
.-. Y^-f Y 2 P 2 + . . . +T_ 1 P_ 1 =y I P 1 +^ f + . . . +y n _,P„_ 1 , 
. Y,P,+Y g P g + . . . + Y. JP.._ 1 _ y 1 P 1 +y,P,+ . . . + // n _ 1 P n „ 1 , 
P + P 2 + . . . +P._ Pa+P.4- . . . t-P._, 

n— 1 n — 13 

where K n _ 1 represents the distance of the centre of pressure 
of P„ P 2 . . . P B _!, from the given plane in the first position, 
and h n _i its distance in the second position. It- distance in 
the first position is therefore the same as in the second. 
Therefore, &c. 

From this proposition, it follows that if a system of weights 
be supported by the resistances of one or more fixed points, 
and if there be any two positions whatever of the weights in 
both of which they are in equilibrium with the resistances 
of those points, then the height of the common centre of 
gravity of the weights is the Bame in both positions. And 
that if there be a Beries of positions in all of which the 
_ tits are in equilibrium about such a resisting point or 
points, then the centre of gravity remains continually at the 
same height as the system passes through this series of posi- 
tions. 

If all these positions of equilibrium be infinitely near tc 



WORK OF PRESSURES. 57 

one another, then it is only during an infinitely small motion 
of the points of application that the centre of gravity ceases 
to ascend or descend; and, conversely, if for an infinitely 
small motion of the points of application the centre of 
gravity ceases to ascend or descend, then in two or more 
positions of the points of application of the system, infi- 
nitely near to one another, it is in equilibrium. 

Work of Pressures applied m different Directions to 
a Body moveable about a fixed Axis. 

62. The work of a pressure applied to a body moveable about 
a fixed axis is the same at whatever point in its proper 
direction that pressure may be applied. 

For let AB represent the direction of a pressure applied 
to a body moveable about a fixed axis 
,s O ; the work done by this pressure 
will be the same whether it be ap- 
plied at A or B. For conceive the 
oody to revolve about O, through an 
exceedingly small angle AOC, or 
BOD, so that the points A and D may describe circular arcs 
AC and BD. Draw Cm, Dn, and OE, perpendiculars to 
AB, then if P represent the pressure applied to AB, P . Am 
will represent the work done by P when applied at A (Art. 
52.), and P . Bn will represent the work done by P when 
applied at B ; therefore the work done by P at A is the same 
as that done by P at B, if Am is equal to Bn. 

Now AC and BD being exceedingly small, they may be 
conceived to be straight lines. Since BD and BE are 
respectively perpendicular to OB and OE, therefore ZDBE 
= / BOE ; "* and because AC and AE are perpendicular to 
OA and OE, therefore Z CAE = Z AOE. Now Am = 

CA . cos. CAE = CA . cos. AOE = ^ . OA . cos. AOE 

OA 

CA 

= OA X 0K Similar1 ^ Bn = DB cos - DBE = DB • cos - 

BOE = 52. OB cos. BOE = ^5 x E, i.e. Am = OE . 

* It is a well-known principle of Geometry, that if two lines be inclined at 
liny angle, and any two others be drawn perpendicular to these, then the inch* 
nation of the last two to one another shall equal that of the first two. 




58 THE ACCUMULATION OF WORK. 

~-t> and Bn = OE ^R. But ^— = - since the / AOC= 
/ BOD, therefore Am = B?i.* 



63. If any number of pressures oe m equiMbriwn about a 
fixed axi*. ihrt, the whole work of those which tend to move 
the system in one direction about that axis is equal to the 
whole work of those which tend to move it in the opposite 
<?<!■• ctiion about the same axis. For let P be any one of such 
a Bystem of pressures, and O a fixed axis, and OM perpen- 
dicular to the direction of P, then whatever may be the 
point of application of P, the work of that pressure is the 
6ame as though it were applied at M. Suppo>e the whole 
T system to be moved through an exceeding small 
/ angle $ about the point O, and let OM be repre- 

V. sented by p, then will pi represent the space 

* v \ described by the point M. which will be actually 
"° in the direction of the force P, therefore the work 
of P=P . p . 4. Xow let P„ P 2 , P 3 , &c. represent those 
pressures which act in the direction of the motion, and P'„ 
P . &c. those which act in the opposite direction, and let 
p^PvPv &C. be the perpendiculars on the first, and p\. />' ._, 
' 3 , &c be the perpendiculars on the second; therefore by 



the pr 



inciple of the equality of moments P, p x + P^ 2 + P 3 ^> s 
+ &c. =: Y\p\ + Y\p\+ F\p' 3 + &c. ; therefore multiplying 
both sides by «, P^ + P 2 jM + P 3 j^ = Y\p\* + P>V + 
P^V + ifec. ; but Pj?A Y\p\t, &c. are the works of the 
forces P n P'„ &c. ; therefore the aggregate work of those 
which tend to move the system in one direction is equal to 
the aggregate of those which tend to move it in the opposite 
direction. 



6i. The accumulation of work in a moving body. 

In every moving body there is accumulated, by the action 
of the forces whence its motion has resulted, a certain 
amount of power which it reproduces upon any resistance 
'Pissed to its motion, and which is measured by the work 
done by it upon that obstacle. Xot to multiply terms, we 
shall speak of this accumulated power of working, thus 
measured by the work it is capable of producing, as accu- 
mulated work. It is in this sense that in a ball fired from 

* Note (o) Ed. App. 



THE ACCUMULATION OF WORK. 59 

a cannon there is understood to be accumulated the work it 
reproduces upon the obstacles which it encounters in its 
flight ; that in the water which flows through the channel 
of a mill is accumulated the work which it yields up to the 
wheel ; * and that in the carriage which is allowed rapidly 
to descend a hill is accumulated the work which carries it a 
considerable distance up the next hill. It is when the pres- 
sure under which any work is done, exceeds the resistance 
opposed to it, that the work is thus accumulated in a moving 
body ; and it will subsequently be shown (Art. 69.) that in 
every case the work accumulated is precisely equal to the 
work done upon the body beyond that necessary to over- 
come the resistances opposed to its motion, a principle 
which might almost indeed be assumed as in itself evident. 

65. The amount of work thus accumulated in a body 
moving with a given velocity, is evidently the same, what- 
ever may have been the circumstances under which its 
velocity has been acquired. "Whether the velocity of a ball 
has been communicated by projection from a steam gun, or 
explosion from a cannon, or by being allowed to fall freely 
from a sufficient height, it matters not to the result ; pro- 
vided the same velocity be communicated to it in all three 
cases, and it be of the same weight, the work accumulated 
in it, estimated by the effect it is capable of producing, is 
evidently the same. 

In like manner, the whole amount of work which it is 
capable of yielding to overcome any resistance is the same, 
whatever may be the nature of that resistance. 

66. To ESTIMATE THE NUMBER OF UNITS OF WORK ACCUMU- 
LATED IN A BODY MOVING WITH A GIVEN VELOCITY. 

Let w be the weight of the body in pounds, and v its 
velocity in feet. 

Now suppose the body to be projected with the velocity v 
in a direction opposite to gravity, it will ascend to the height 
h from which it must have fallen, to acquire that same velo- 
city v (Art. 47.); there must then at the instant of projection 
have been accumulated in it an amount of work sufficient to 
raise it to this height A; but the number of units of work 

* This remark applies more particularly to the under-shot wheel, which il 
carried round by the rush of the water. 



60 



THE ACCUMULATION OF WORK. 



requisite to raise a weight w to a height h, is represented by 

wh\ this then is the number of units of work accumulated 
in the body at the instant of projection. But since h is the 
height through which the body must fall to acquire the velo- 

city v, therefore v' i —2gh (Art. 47.); therefore h=%— ; whence 

it follows that if U represent the number of units of work 
accumulated, 

U=^V (44). 

Moreover it appears by the last article that this expression 
represents the work accumulated in a body weighing w 
pounds, and moving with a velocity of v feet, whatever may 
nave been the circumstances under which that velocity was 
accumulated. 

The product (— Jv* is called the vis viva of the body, so 

that the accumulated work is represented by half the vis 

viva, the quotient ( — j is called the mass of the body.* 

67. To estimate the work accumulated in a body, or lost by 
it, as it passes from one velocity to another. 

In a body whose weight is w, and which moves with a 

velocity v there is accumulated a number of units of work 

w 
represented (Art. 66.) by the formula J— 1\ After it has 

passed from this velocity to another Y, there will be accumu- 

w 
lated in it a number of units of work, represented by -J— Y 2 , 

so that if its last velocity be greater than the first, there 

will have been added to the work accumulated in it a num- 

w w 

ber of units represented by -£— Y 2 — J— 1> 2 ; or if the second 

velocity be less than the first, there will have been taken 

from the work accumulated in it a number of units repre- 

w w 

sented by £— v 1 — J— Y 2 . So that generally if U represent 

the work accumulated or lost by the body, in passing from 
the velocity v to the velocity Y, then 

* Note (p) Ed. App. 




THE ACCUMULATION OF WORK. 61 



U=±ij{V>-v>] (45), 

where the ± sign is to be taken according as the motion is 
accelerated or retarded. 



3. The work accumulated in a hod]/, whose motion is accele- 
rated through any given space hy given forces is equal to 
the work which it would he necessary to do upon the hody 
to cause it to move hack again through the same space 
when acted upon hy the same forces. 

For it is evident that if with the velocity which a body 
has acquired through any space AB by the 
action of any forces whose direction is from A 
towards B, it be projected back again from B 
towards A, then as it returns through each 
successive small part or element of its path, it 
will be retarded by precisely the same forces as those by 
which it was accelerated when it hefore passed through it ; 
so that it will, in returning through each such element, lose 
the same portion of its velocity as before it gained there ; 
and when at length it has traversed the whole distance BA, 
and reached the point A, it will have lost between B and A 
a velocity, and therefore an amount of work (Art. 67.), 
precisely equal to that which before it gained between A 
and B. Now the work lost between B and A is the work 
necessary to overcome the resistances opposed to the motion 
through BA. The work accumulated from A to B is there- 
fore equal to the work which would be necessary to over- 
come the resistances between B and A, or which would be 
necessary to move the body from a state of rest, and with a 
uniform motion, in opposition to these resistances, through 
BA. Let this work be represented by U ; also let v be the 
velocity with which the body started from A, and V that 

W 

which it has acquired at B. Then will £ — (Y 2 — v*) repre- 
sent the work accumulated between A and B, 

, ^(vw)=u, , v-sJ^. 

If the body, instead of being accelerated, had been 
retarded, then the work lost being that expended in over- 
coming the retarding forces, is evidently that necessary to 



03 THE ACCUMULATION OF WORK. 

move the body uniformly in opposition to these retarding 
forces through AB; so that if this force be represented by 

IT, then, since i — (V — V a ) is in this case the work lost, we 

if 

shall have v % — Y*= -|^-. Therefore, generally, 

*W=±^ (46), 

where the sign ± is to be taken according as the motion is 
accelerated or retarded. 



69. The work accumulated in a "body which has moved 
through any space acted upon by any force, is equal to the 
excess of the work which has been done upon it by those 
forces which tend to accelerate its motion above that which 
has been done upon it by those which tend to retard its 
motion. 

For let R be the single force which would at any point P 
(see last fig.) be necessary to move the body back again 
through an exceeding small element of the same path (the 
other forces impressed upon it remaining as before) ; then it 
follows by Art. 54. that the work of R over this element of 
the path is equal to the excess of the work over that 
element of the forces which are impressed upon the body in 
the direction of its motion above the work of those 
impressed in the opposite direction. Now this is true at 
every point of the path ; therefore the whole work of the 
force E. necessary to move the body back again from B to A 
is equal to the excess of the work done upon it, by the 
impressed forces in the direction of its motion, above the 
work done upon it by them in a direction opposed to its 
motion ; whence also it follows, by the last proposition, that 
the accumulated work is equal to this excess. There- 
fore, &c. 

*70. If P represent the force in the direction of the 
motion which at a given distance S, measured along the 
path, acts to accelerate the motion of the body, this force 
being understood not to be counteracted by any other, or tc 
be the surplus force in the direction of the motion over and 



THE ACCUMULATION OF WORK. 63 

above any resistance opposed to it, then will / FdS be the 

o 

work which must be done in an opposite direction to over- 

s 

come this force through the space S, or U— fPd$, 

o 

s 

2gf¥dS 
:. by equation (46), "V— u a =± — ~ (47). 

71. If the force P tends at first towards the direction in 
which the body moves, so as to accelerate the motion, and 
if after a certain space has been described it changes its 
direction so as to retard the motion, and U, represent the 
value of U in respect to the former motion, and "V, the 
velocity acquired when that motion has terminated, whilst 
U 2 is the value of U in respect to the second or retarded 
motion, and if v be the initial and V the ultimate or actual 
velocity, then 

v i v — ^ » 

y Vl_ W ' 
••• Y'-- 2 ^ (43). 

As U 2 increases, the actual velocity V of the body con- 
tinually diminishes ; and when at length XJ,=U 1 , that is 
when the whole work done (above the resistances) in a 
direction opposite to the motion, comes to equal that done, 
before, in the direction of the motion, then V=v, or the 
velocity of the body returns again to that which it had 
when the force P began to act upon it. This is that gene- 
ral case of reciprocating motion which is so frequently pre- 
sented in the combinations of machinery, and of which the 
crank motion is a remarkable example. 

*72. If the force which accelerates the body's motion act 
always towards the same centre S, and Sb be taken equal to 




64: THE ACCUMULATION OF WORK.. 

SB, it has been shown (Art 55.) that the work 
necessary to move the body along the curve from 
B to A, is equal to that which would be necessary 
to move it through the straight line bA. The 
accumulated work is therefore equal to that neces- 
sary to move the body through the difference bA 
of the two distances SA and SB (Art. 68.). If these 
distances be represented by It, and R 2 , and P 
represent the pressure with which the body's motion along 
bA would be resisted at any distance R from the Joint S, 

Ri 

then / FdH will represent this work. Moreover the work 

accumulated in the body between A and B is represented 
by i— (Y 2 — v\ if V represent the velocity at B and v that 
at A, 



a «/ 



P^R 



Ro 



.-. V— ^=^J¥dR (49). 



73. The work accumulated in the body while it descends 
the curve AB, is the same as that which it would acquire in 
falling directly towards S through the distance Ab, for both 
of these are equal to the work which would be necessary to 
raise the body from b to A. Since then the work accumu- 
lated by the body through AB is equal to that which it would 
accumulate if it fell through Ab, it follows that velocity 
acquired by it in falling, from rest, through AB is equal to 
that which it would acquire in falling through A b. For if 
Y represent the velocity acquired in the one case, and V, 
that in the other, then the accumulated work in the nrst case 

is represented by -J — Y 2 , and that in the second case by -J — Y x *, 

W ~^Y 9 

therefore i — Y 2 = £ — Y 2 therefore Y=Y 1 . 

- g 2 g li 

From this it follows, that if a body descend, being pro- 
jected obliquely into free space,, or sliding from rest upon 
any curved surface or inclined plane, and be acted upon only 
by the force of gravity (that is, subject to no friction or 
resistance of the air or other retarding cause), then the velo- 



THE ACCUMULATION OF WORK. 65 

city acquired by it in its descent is precisely the same as 
though it had fallen vertically through the same height. 

74. Deflxltiox. The angular velocity of a body which 
rotates about a fixed axis is the arc which every particle of 
the body situated at a distance unity from the axis describes 
in a second of time, if the body revolves uniformly / or, if 
the body moves with a variable motion, it is the arc which it 
would describe in a second of time if (from the instant when 
its angular velocity is measured) its revolution were to 
become uniform. 

75. The accumulatiox of work lx a body which 
rotates about a fixed axis. 

Propositions 68 and 69 apply to every case of the motion 
of a heavy body. In every such case the work accumulated 
or lost by the action of any moving force or pressure, whilst 
the body passes from any one position to another, is equal 
to the work which must be done in an opposite direction, to 
cause it to pass back from the second position into the first. 
Let us suppose U to represent this work in respect to a body 
of any given dimensions, which has rotated about a fixed 
axis from one given position into another, by the action of 
given forces. 

Let a be taken to represent the angular velocity of the 
body after it has passed from one of these positions into 
another. Then since a is the actual velocity of a particle at 
distance unity from the axis, therefore the velocity of a par- 
ticle at any other distance p 1 from the axis is ap i; Let m- 
represent the weight of each unit of the volume of the body, 
and m, the volume of any particle whose distance from the 
axis is p„ then will the weight of that particle be y>m x ; also 
its velocity has been shown to be ap : , therefore the amount 
of work accumulated in that particle is represented by 

4— a Pi 3 or by K-^iP, - 

Similarly the different amounts of work accumulated in 
the other particles or elements of the body whose distances 
from the axis are represented by p 2 , p 3 , . . . and their 

volumes by m 2 , ra 3 , m A . . . ., are represented by J^-w^p,", 
fjt, • ^ 

^a a -?/^p 3 2 5 &c. ; so that the whole work accumulated is repre- 
y 

5 



06 ANGULAR VELOCITY. 



U> tL U. 

eented by the sum i« 9 -m 1 p 1 '+ i« 2 -?ft 2 p a a + £ aa -^,p/ + 
sf jf y 

> or byK- Kp 1 3 +^ 2 p, a +^3p, 2 + |. 

The sum ra^H- m a p 2 2 + m 3 p 3 a + . . . ., or 2rap 2 taken in 



respect to all the particles or elements which compose the 
body, is called its moment of inertia in respect to the 
particular axis about which the rotation takes place. Let it 

be represented by I; then will Ja 2 . ( — J . I, represent the 

whole amount of work accumulated in the body whilst it has 
been made to acquire the angular velocity a from rest. If 
therefore U represent the wort which must be done in an 
opposite direction to cause the body to pass back from its 
last position into its first, 






If instead of the body's first position being one of rest, it 
had in its first position been moving with an angular velocity 
a, which had passed, in its second position, into a velocity 
a ; and if IT represent, as before, the work which must be 
done in an opposite direction, to bring this body back from 

its second into its first position, then is £« 2 ( '-) I — J«/ ( -J I, 

- ) (a 2 — a^) I, the work accumulated between the first 
and second positions ; therefore 

*(fV- a '') i=±ij ' 

.-. rf=^±»(j)§ (51), 

where the sign ± is to be taken according as the motion is 
accelerated or retarded between the first and second posi- 
tions, since in the one case the angular velocity increases 
during the motion, so that a 2 is greater than a x 9 , whilst in the 
latter case it diminishes, so that a 2 is less than a^. 

76. If during one part of the motion, the work of the 



ANGULAR VELOCITY. 67 

impressed forces tends to accelerate, and during another to 
retard it, and the work in the former case be represented by 
TJj, and in the latter by U„ then 

,.•=.,•« (f)&=&> m 

From this equation it follows that when 112=11,, or when 
the work U 2 done by the forces which tend to resist the 
motion at length, equals that done by the forces which tend 
to accelerate the motion, then a= a,, or the revolving body 
then returns again to the angular velocity from which it set 
out. Whilst, if TJ 2 never becomes equal to U 1 in the course 
of a revolution, then the angular velocity a does not return 
to its original value, but is increased at each revolution ; 
and on the other hand, if U 2 becomes at each revolution 
greater than IT,, then the angular velocity is at each revolu- 
tion diminished. 

The greater the moment of inertia I of the revolving 
mass, and the greater the weight ^ of its unit of volume 
(that is, the heavier the material of which it is formed), the 
less is the variation produced in the angular velocity a by 
any given variation of U or TJ l — U 2 at different periods of 
the same revolution, or from revolution to revolution ; that 
is, the more steady is the motion produced by any variable 
action of the impelling force. It is on this principle that 
the fly-wheel is used to equalize the motion of machinery 
under a variable operation of the moving power, or of the 
resistance. It is simply a contrivance tor increasing the 
moment of inertia of the revolving mass, and thereby 
giving steadiness to its revolution, under the operation of 
variable impelling forces, on the principles stated above. 
This great moment of inertia is given to the fly-wheel, by 
collecting the greater part of its material on the rim, or 
about the circumference of the wheel, so that the distance 
p of each particle which composes it, from the axis about 
which 'it revolves, may be the greatest possible, and thus 
the sum 2mp 3 , or I, may be the greatest possible. At the 
same time the greatest value is given to the quantity m-, by 
constructing the wheel of the heaviest material applicable 
to the purpose. 

What has here been said will best be understood in its 
application to the crank. 



68 ANGULAR VELOCITY. 

77. If we conceive a constant pressure Q to act upon the 
B arm CB of the crank 

..- "V/^^^^q m tne direction AB of 

/ //*>)>. ^^"^S^L tne crank rod, and a 

i c ($y : T ^^^ s _^_-^_- cons tant resistance R 

/'y R to be opposed to the 

revolution of the axis 
C always at the same perpendicular distance from that axis, 
it is evident that since the perpendicular distance at which 
Q acts from the axis is continually varying (being at one 
time nothing, and at another equal to the whole length CB 
of the arm of the crank), the effective pressure upon the 
arm CB must at certain periods of each revolution exceed the 
constant resistance opposed to the motion of that arm, and 
at other periods fall short of it ; so that the resultant of 
this pressure and this resistance, or the unbalanced pressure 
P upon the arm, must at one period of each revolution have 
its direction in the direction of the motion, and at another 
time opposite to it. Representing the work done upon the 
arm in the one case by Uj, and in the other by TJ 9 , it follows 
that if U 1 =U 3 the arm will return in the course of each 
revolution, from the velocity which it had when the work 
Uj began to be done, to that velocity again when the work 
U 2 is completed. If on the contrary IT, exceed U 2 , then the 
velocity will increase at each revolution ; and if U, be less 
than u 3 , it will diminish. It is evident from equation (52), 
that the greater the moment of inertia I of the body put 
in motion, and the greater the weight ^ of its unit of 
volume, the less is the variation in the value of a, produced 
by any given variation in the value of TJ, — U 2 ; the less 
therefore is the variation in the rotation of the arm of the 
crank, and of the machine to which it gives motion, pro- 
duced by the varying action of the forces impressed upon it. 
Now the fly-wheel being fixed upon the same axis with the 
crank arm, and revolving with it, adds its own moment of 
inertia to that of the rest of the revolving mass, thereby 
increasing greatly the value of I, and therefore, on the prin- 
ciples stated above, equalizing the motion, whilst it does not 
otherwise increase the resistance to be overcome, than by 
the friction of its axis, and the resistance which the air 
opposes to its revolution.* 

* We shall hereafter treat fully of the crank and fly-wheel. 



ANGULAR VELOCITY. 



78. The rotation of a body about a fixed axis when acted 
upon by no other moving force than its weight. 

Let U represent the work necessary to raise it from its 
second position into the first if it be descending, or from its 
first into its second position if it be ascending, and let a x be 
its angular velocity in the first position, and a in the second ; 
then by equation (51), 

■•=■•«©(?)■ 

Now it has been shown (Art. 60.), that the work necessary 
to raise the body from its second position into the first if it 
be descending, or from its first into its second if it be 
ascending (its weight being the only force to be overcome), 
is the same as would be necessary to raise its whole weight 
collected in its centre of gravity from the one position into 
the other position of its centre of gravity. Let OA repre- 
sent the one, and CA X the other position of 
rx? the body, and G and G- x the two correspond- 

~\^-- % ing positions of the centre of gravity, then 

^^j^p^*...^ 1 will the work necessary to raise the body 

J.jSb\ from its position CA to its position CA l5 be 

5 xj* equal to that which is necessary to raise its 

whole weight W, supposed collected in G, 

from that point to G x ; which by Article 56, is the same as 

that necessary to raise it through the vertical height GM. 

Let now CG=CG,=:A, let CD be a vertical line through 
C, let G 1 CD=^ 1 and GCD=d, in the case in which the 
body descends, and conversely when it ascends; therefore 
G'M='N'N l = CN—CN l =h cos. & — h cos. 6 l when the body 
descends, or =h cos. 6 X — h cos. 6 when it ascends from the 
position AC to AC 15 since in this last case GCD=^ and 
G^D^. Therefore GM=±h (cos. & — cos. d,), the sign ± 
being taken according as the body ascends or descends. 
Now U=W . GK=±Wh (cos. &— cos. a,), 

,\ by equation (51) a 9 =a l a + i — z— \ (cos. 6 — cos. ^). 

If M represent the volume of the revolving body Mm-=W, 

.-. a * = «; + p™) (cos. 6— COS. a,) (53). 

When the body has descended into the vertical position, 



70 MOMENT OF INERTIA. 

6=0, so that (cos. d — cos. ^)=1 — cos. ^=2 sin. 2 ^. When 
it has ascended into that position d=<r, so that (cos. d — cos. 
«,)=— (1+cos. &>)=— 2 cos. 2 ^. 
In the first case, therefore, 

a > =<+ (^y in , ¥ ^ 

In the second case, 

a , =a ,_p^ cos ^ (55) _ 

When the body has descended or ascended into the hori- 
zontal position 0=~> s0 tnat ( cos - & — cos - & i) = — cos. 6 im But 
it is to be observed, that if the body have descended into 



the horizontal position, 6 t must have been greater than =-, 

and therefore cos. 6 t must be negative and equal to — cos. 
BCG, ; so that if we suppose ^ to be measured from CB or 
CD, according as the body descends or ascends, then (cos. 
— cos. ^)=±cos. d„ and we have for this case of descent 
or ascent to a horizontal position 

a , =a _, ± %AM og _ t (66) 

If the body descend from a state of rest, a i= :0. 
/. by equation (53) a 2 — g ^ (cos. 6 — cos. 6 t ) . . . (57). 

Thus the angular velocity acquired from rest is less as the 
moment of inertia I is greater as compared with the volume 
M, or as the mass of the body is collected farther from its 
axis. 



The Moment of Inertia. 

79. Having given the moment of inertia of a body, or system 
of bodies, about an axis passing through its centre of 
gravity, to find its moment of inertia about an axis, par- 
allel to the first, passing through any other point in the 
body or system. 

Let m, be any element of the body or system, 7n Y AGr a 



MOMENT OF DfERTIA. 



71 



mi plane perpendicular to the axis, about 

/\a which the moments are to be measured, A 

/ ,/X. the point where this plane is intersected 

^ / : ; ;'_"'_ _\ by that axis, and G the point where it is 

intersected by the parallel axis passing 

through the centre of gravity of the body. Join AG, 

Aw l5 Gm,, and draw m^I, perpendicular to AG. Let 

Am 1 =p 1 , Gm 1 =r 1 , GtM 1 =x 1 , AG =h. 

Now (Euclid, 2—12.), Am, 2 = AG ? +Gm 1 '+ 2 AG . GM„ 

or tfszh'+rf+Sitor 
If therefore the volume of the element be represented by 
m l5 and both sides of the above equation be multiplied by it, 
P*m l = lbm x + r*m 1 + 2 Aa^m^ 
And if ra 2 , m t , ra 4 , &c. represent the volumes of any other 
elements, and p 2 , r„ a? 2 ; p 3 , r„ a? 3 , &c. be similarly taken in 
respect to those elements, then, 

p 2 2 ra 2 = A 2 m 2 + r*m 2 + 2 Aa? 2 m 2 , 

p 3 2 m 3 = A 2 m 3 + r, s ra, + 2 A# 3 ra 3 , 

&c.=&c. 

Adding these equations we have, p, 2 ?^ + p 2 2 m 2 + p 3 2 ra 3 + . . . 

=#(ra 1 4-w& a +w& 3 + • • • • ) + (r 1 2 m 1 +rsm,+r*m z + ....)+ 

2A(a? 1 m 1 +-a? a in a +aj,m,4- • • • )> 

or 2p 2 m=A 2 2m+27 >2 m+2A2a?m. 

Now 2ccm is the sum of the moments of all the elements 
of the body about a plane perpendicular to AG, and passing- 
through the centre of gravity G of the body. Therefore 
(Art. IT.) 2#m=0, 

Also 2p 2 m is the moment of inertia of the body about the 
given axis passing through A, and 2r 2 m is the moment of 
inertia about an axis parallel to this, passing through the 
centre of gravity of the body. Let the former moment be 
represented by I, ; and the latter by I ; and let the volume 
of the body 2m be represented by M, 

/. I^tfM+I (58). 

From which relation the moment of inertia (I,) about any 
axis may be found, that (I) about an axis parallel to it, and 

Eassing through the centre of gravity of the body being 
mown. 

80. The radius of gyration. If we suppose /', to be the 
distance from the axis passing through A, at which distance, 



72 MOMENT OF INERTIA. 

if the whole mass of the "body were collected, the moment of 
inertia would remain the Bame, bo that jfe^MssI,, then Tc x is 
called the radius of gyration, in respect to that axis. 

If h be the radius of gyration, similarly taken in respect 
to the axis passing through G„ so that #M=I, then, substi 
tilting in the preceding equation, and dividing by M, 

k:=K + k* (59). 

The following are examples of the determination of the 
moments of inertia of bodies of some of the more common 
geometrical forms, about the axes passing through their cen- 
tres of gravity : they may thence be found about any other 
axes parallel to these, by equation (58). 



•81. The moment of inertia of a thin uniform rod about an 
axis perpendicular to its length and passing through its 
middle point. 

Let m represent an element of the rod contained between 

A | two plane sections perpendicular to its 

c < I * fr faces, the area of each of which is k, and 

t At m n whose distance from one another is Ap, 

I and let n and Ap be so small that every 

point in this element may be considered to be at the same 
distance p from the axis A, about which the rod revolves. 
Then is the volume of the element represented by /cAp, and 
its moment of inertia about A by «;p 2 Ap. So that the whole 
moment of inertia I of the bar is represented by 2«p 3 Ap, or, 
since k is the same throughout (the bar being uniform), by 
zc2p 3 Ap ; or since Ap is infinitely small, it is represented by 

the definite integral k I pVZp, where I is the whole length 
of the bar, 

orI =T V«? (60). 



*82. The moment of inertia of a thin rectangular lamina 
about an axis, passing through its centre of gravity, and 
parallel to one of its sides. 

It is evident that such a lamina may be conceived to be 



MOMENT OF INERTIA. 73 

made up of an infinite number of slender 
i rectangular rods of equal length, each of 

z::::::^:_::::::^ ^:^ which will be bisected by the axis AB, 
-— -I ^^^ ^ at ^ moment of inertia of the 
whole lamina is equal to the sum of the 
moments of inertia of these rods. Xow if a be the section 
of any rod, and I the length of the lamina, then the moment 
of inertia of that rod is, by the last proposition, represented 
by T \d 3 ; so that if the section of each rod be the same, and 
they be n in number, then the whole moment of inertia of 
the lamina is T \nid\ Xow nn is the area of the transverse 
section of the lamina, which may be represented by K, so 
that the moment of inertia of the lamina about the axis AB 
is represented by the formula 

I=*K* (61)- 



*83. The moment of inertia of a rectangular parallelopipe- 
don about an axis, jjassing through its centre of gravity, 
and parallel to either of its edges. 

Let CD be a rectangular parallelopipedon, and AB an 
axis passing through its centre of gravity and 
parallel to either of its edges ; also let ah be 
an axis parallel to the first, passing through 
the centre of gravity of a lamina contained 
by planes parallel to "either of the faces of the 
paralielopiped. Let a, b, c, represent the 
three edges ED, EF, EG, of the paralielo- 
piped, then will the moment of inertia of the lamina about 
the axis ab be represented by T l jKb% where K is the trans- 
verse section of the lamina "(equation 61). Now let the 
perpendicular distance between the two axes AB and db be 
represented by x. Then (by equation 58) the moment of 
inertia of the lamina about the axis AB is represented by 
the formula a^M + rVK^A where M represents the volume of 
the lamina. Let the thickness of the lamina be represented 
by &x ; .*. M — ab^x, K — a&x ; .*. m 1 in a of lam a = absfAx -f 
r ^ab z Ax ; /. whole m* in a of paralielopiped == ab^x^x + 
fjab^&x; or taking Ax infinitely small, and representing the 
moment of inertia of the paralielopiped by I. 




\—abl x*dx + T \ ab* f dx ; 



-\e 



74 MOMENT OF INERTIA. 

or ^ablUW-K-icyi + A^ftoM-WJ 

= T \abc' + T ^ab 3 c, 

;.I=z^abo{b % +<f) (62). 



^S-i. The moment of inertia of an upright tria/ngvlar prisr.i 

about a vertical axis passing through its centre of ' g rarity. 

Let AB be a vertical axis passing through the centre of 
gravity of a prism, whose horizontal section ia 
an isosceles triangle having the equal sides ED 
7 and EF. 

Let two planes be drawn parallel to the face 
J DF of the prism, and containing between them 
a thin lamina pq of its volume. Let Cm, the 
perpendicular distance of an axis passing through 
the centre of gravity of this lamina from the 
axis AB, be represented by x\ also let &x represent the 
thickness of the lamina. 

Let DF= a, DG = b. and let the perpendicular from the 
vertex E to the base DF of the triangle DEF be represented 

bye, 

/. EC = fc, Era =-£<?, — x ; also f^ = , 

Dr c 

:.pq = - ($c—x) ; also transverse section K of lamina = b±x. 
c 

:. volume M of lamina = (f c— x)&x. Therefore by equa- 

c 

tions (58) and (61), 

m* in* of lam a about AB=^{%c—xy±x + T \b a ,($c-x) 9 Ax-, 

c c 

:. m* in a of prism about 

ah p + ic ba* /%+iii 

AB = — l(%c-xyax + T \— f(%c—aydx. 
c d -*c & *> -4c 

Performing the integrations here indicated, and represent- 
ing the inertia of the prism about AB by I, we have 

I= T \abc(±a' + ^) (63). 



MOMENT OF INERTIA, 



75 



*85. The moment of inertia of a solid cylinder about its 
axis of symmetry. 

Let AB be the axis of such a cylinder, whose radius AC 
is represented by a, and its height by b. Con- 
ceive the cylinder to be made up of cylindrical 
rings having the same axis ; let AP= p be the 
internal radins of one of these, and let its thick- 
ness PQ be represented by Ap, so that p + Apis 
the exteral radius AQ of the ring. Then will 
the volume of the ring be represented by 
tf£(p-f Ap) 2 — *#p 2 , or by <:r&[2pAp + (Ap) 2 ] ; or if Ap 
be taken exceedingly small, so that (Ap) 2 may vanish as com- 
pared with 2pAp ? then is the volume of the ring represented 
by 2*5pAp. 

JSTow this being the case, the ring may be considered as an 
element AM of the volume of the solid, every part of which 
element is at the same distance p from the axis AB, so that 
the whole moment of inertia 2p 2 AM of the cylinder = 

2p*(2^pAp) = 2^2p 3 Ap, 




=2*b f fd?=&<ba K 



(64). 



*86. The moment of inertia of a hollow cylinder about its 
axis of symmetry. 

Let a x be the external radius AC, and « 2 the internal 
radius AP, and b the height of the cylinder ; 
then by the last proposition the moment of in- 
ertia of the cylinder CD, if it were solid, would 
be farba* ; aiso the moment of inertia of the 
cylinder PR, which is taken from this solid to 
^v* form the hollow cylinder, would be ^ba*. Now 
i B let I represent the moment of inertia of the hol- 
low cylinder CP, therefore I-\-^ba' 2 i =^ba l % 

;.I=^b(a;~a:)=^b{a:-a^)(a; + a;)=i^b(a—a i ) 
(a^a^a' + a*). 

Let the thickness a x — a 2 of the hollow cylinder be repre- 
sented by c, and its mean radius \{a v + a^ by R, therefore 




76 



s r OF IXEKTIA. 



Substituting these values in the preceding equation, we ol» 
tain 

l-i- i: i:- ■ ■ ■-; 

v 7. The moment of inertia of a cr 

; through Us centre of gra/ovty^ an alar 

try. 

Let AB be such an axis, and let PQ represent a lamina 
contained between planes perpendicular to 
this axis, and exceedingly near to each other. 
Let CD, the axis of the cylinder, be repre- 
sented by I, its radius by a. and let CM=^. 
Take Xv to represent the thickness of the 
lamina, and let MP=y. Now this lamina 
maybe considered a rectangular parallelo 
piped traversed through its centre ot' gravity by the axis AB ; 
therefore by equation (62) its moment of inertia about that axis 
is represented by M A ' ' - - =ib\by+4tf\Ax. 

Xow the whole moment of inertia I of the cylinder about 
AB is evidently equal to the sum of the moments of inertia 
of all such lamina 1 : 

;. I=ihi \Vy+4*f] Xc^lf^y + iy'yl.c. 

— a 

Also, since x and y are the co-ordinates of a point in a 
circle from its centre, therefore y= (V— .r i 1 . Substituting 
this value of y, and integrating according to the well known 
rules of the integral calculus,* we have 




I=i*oa\a % +W) 



(66). 



*SS. The moment of inertia of a cone about its axis of 
symmetry. 
I 

The cone may be supposed to be made up of lamina?, such 
as PQ, contained by planes perpendicular to 
the axis of symmetry AB, and each having its 
centre of gravity in that axis. Let BP=./, and 
let -Vr represent the thickness of the lamina, 
and y its radius PR. Then, since it may be 
considered a cylinder of very small height, its 
moment of inertia about Al3 (equation 6±) is 
represented by ±-y'\r. Xow the moment of 

Church's Diff. and Intcg. Calculus, Arts. 148, 149. 




MOMENT OF INERTIA. , 77 

inertia I of the whole cone is equal to the sum of the mo- 
ments of all such elements, 

Let the radius of the base of the cone be represented by 

x b 7) 

/z, and its height by b ; therefore- =-, therefore Ax= -Ay ; 

b b P a 

.". 1= i*a* yiAy=i *aJ y * dy ' 

o 

:.I= Y \«oa* (67). 



89. The moment of inertia of a sphere about one of its 
diameters. 

Let G be the centre of the sphere and AB the diameter 
about which its moment is to be determined. 
Let PQ be any lamina contained by planes 
perpendicular to AB ; let CM = x, and letAa? 
P 8 represent the thickness of the lamina, and y its 
radius ; also let CA=a ; then since this lamina, 
being exceedingly thin, may be considered a 
cylinder, its moment of inertia about the axis AB is (equa- 
tion 64) ^y*Ax ; and the moment of inertia I of the whole 
sphere is the sum of the moments of all such laminae, 




I=%f2y i Ax=i* I y 4 dx. 



Now by the equation to the circle y^=za t — a? 2 , therefore 
y*=a*—2a i x 2 -}-x\ If this value be substituted for y% and 
the integration be completed according to the common 
methods, we shall obtain the equation, 

I=iW (68). 



90. The moment of inertia of a cone about an axis passing 
through its centre of gra/uity and perpendicular to its axis 
of symmetry. 

Let CD be an axis passing through the centre of gravity 




T8 KOMEHT OF INERTIA. 

G of the cone, and perpendicular to its axis of 
symmetry, and let GP the distance of the lamina 
from G, measured along the axis, be represented 

by x : also let the thickness of the lamina be re- 
presented by&x. Now this lamina may be con- 
sidered a cylinder of exceedingly small thick- 
ness. It* its radius be represented by //. its mo- 
ment of inertia about an axis parallel to CD passing through 
its centre, is therefore (equation 66) represented bv 
■~ &+¥*&)*]*&, or tf Ax De assumed exceedingly small, 
it is represented by \~y*Xc. Xow this being the moment of 
the lamina about an axis parallel to CD, passing through its 
centre of gravity, and the distance of this axis from CD be- 
ing Xj and also the volume of the lamina being ^/'-X>\ it fol- 
lows (equation 58), that the moment of the lamina about CD 
is represented by cyVAz + ^ry 4 ^— ar{yV+ \i/\ \±x. 

Xow the moment I of the whole cone about CD equals 
the sum of the moments of all such elements, 

Xow if a be the radius of the base of the cone and b its 
height, then since BGr=£5, 

a 

:.l^^a'b\a'+ib'\ (69). 



91. The moment of inertia of a segment of a sphere about 
a diameter parallel to the plane of section. 

Let ADBE represent any such portion of a sphere, and 
AB a diameter parallel to the plane of section. 
Let CD=a, CE=5. and let PQ be any lamina 
contained by planes parallel to the plane of 
section: let the distance of the lamina from 
C=a\ and let its thickness be ^x and its radius 
y. Then considering it a cylinder of exceeding small thick- 




MOVING FORCES. 79 

ness, its moment of inertia about an axis passing through its 
centre of gravity and parallel to AB, is represented (equa- 
tion 66) by i^?/ 2 {y 2 +3-( A ») 2 | A ^, or (neglecting powers of &x 
above the" first by \ty*Ax. Hence, therefore, the moment of 
this lamina about the axis AB is represented (equation 
58) by ^(Aajjar'-f Jiry*Aaj, or by ^jyV+iy 4 J a#; now the 
whole moment I of inertia of ADBE about AB is evidently 
equal to the sum of the moments of all such laminae, 

/. I=*2 {y V +i^ ±X = «ftfx'+\if)dx. 

-b 

Now f=a*-x% therefore ^+ly 4 =i{2aV— 3a>* + a*}. 
Substituting this value in the integral and integrating, we 
have 

I=^{16a i + loa i b.+10a 2 h 5 -9b 5 \* (TO) 



THE ACCELEEATIOX OF MOTION BY GIVEN 
MOVING FOKCES. 

92. If the forces applied to a moving body in the direc- 
tion of its motion exceed those applied to it in the opposite 
direction (both sets of forces being resolved in the direction 
of a tangent to its path), the motion of the body will be ac- 
celerated ; if they fall short of those applied in the opposite 
direction, the motion will be retarded. In either case the 
excess of the one set of forces above the other is called the 
moving force upon the body : it is measured by that single 
pressure which being applied to the body in a direction op- 
posite to the greater force, would just balance it ; or which, 
had it been applied to the body (together with the other 
forces impressed upon it) when in a state of rest, would have 
maintained it in that state ; and which, therefore, if applied 
when its motion had commenced, would have caused it to 
pass from a state of variable to one of uniform motion. Thus 
the moving force upon a body which descends freely by gra- 
vity, is measured by its weight, that is, by the single force 
which, being applied to the body before its motion had com- 
menced in a direction opposite to gravity, would just have 
supported it, and which being applied to it at any instant of 

* Note (?) Ed. App. 



SO RELATIONS OF 

it- descent, would have caused its motion at that instant to 
- ate of variable to a Btate of uniform motion. 
If the resistance of the air upon ir> descent be taken into 
account, then the movingforce upon the body at any instant 
is measured by that single pressure which, being applied up- 
ward-, would, together with tin- resistance of the air at that 
instant, just balance the weight of the body. 

A moving force being thus understood to be measured by 
i\j>r- 88it r> -," being in fact the unbalanced pressure upon the 
moving body, the following relations between the amount of 
a moving force thus measured, and the degree of acceleration 
produced by it will become intelligible. These are laws of 
motion which have become known by experiment upon the 
motions of the bodies immediately around us, and by obser- 
vation upon those of the planets. 

93. "When the moving force upon a body remains con- 
stantly the same in amount I as measured by the equivalent 
pressure) throughout the motion, or is a uniform moving 
force, it communicates to it equal additions of velocity in 
equal successive intervals of time. Thus the moving force 
upon a body descending freely by gravity (measured by its 
weight) being constantly the same in amount throughout its 
descent (the resistance of the air being neglected i, the body 
receives from it equal additions of velocity in equal succes- 
sive intervals of time, viz. 32} feet in each successive second 
of time (Art. 44.). 

9-i. The increments of velocity communicated to equal 
bodies by unequal moving forces (supposed uniform as above) 
are to one another as the amounts of those moving forces 
(measured by their equivalent pressure 

Thus let P and P l be any two unequal moving forces upon 
two equal bodies, and let them act in the directions in which 
the bodies respectively move : let them be the only forces 
tending to communicate motion to those bodies, and remain 
constantly the same in amount throughout the motion. Also 
let f and r, represent the additional velocities which these 
two forces respectively communicate to those two equal 
bodies in each successive second of time ; then it is a law of 
the motion of bodies, determined bv observation and experi- 
ment, that P : P, ::/:/;. 

* Pressure and moving force are indeed but different modes of the operation 
of the same principle of force. 



PKESSURE AOT) MOTION. 81 

If one of the moving forces, as for instance P„ be the 
weight W of the body moved, then the value f of the 
increment of velocity per second corresponding to that 
moving force is 32 1 (Art. 44.) represented by g, 

W 
•••P=-/ (71). 



95. If the amonnt or magnitude of the moving force does 
not remain the same throughout the motion, or if it be a 
variable moving force, then the increments of velocity com- 
municated by it in equal successive intervals of time are not 
equal; they increase continually if the moving force 
increases, and they diminish if it diminishes. 

If two unequal moving forces, one or both of them, thus 
variable in magnitude, become the moving forces of two 
equal bodies, the additional velocities which they would 
communicate in the same interval of time to those bodies, 
if at any period of the motion from variable they become 
■uniform, are to one another (Art. 94.) as the respective 
moving forces at that period of the motion. 

Thus let/" and f represent the additional velocities which 
would thus be communicated to two equal bodies in one 
second of time, if at any instant the pressures P and P l5 
which are at that instant the moving forces of those bodies, 
were from variable to become constant pressures, then 
(Art. 94.), 

PrP,::/:/, 

This being true of any two moving forces, is evidently true, 
if one of them become a constant force. Let P, represent 
the weight W of the body, then will f be represented 

:.F:W::f:g. 

Let the moving force P be supposed to remain constant 
during a number of seconds or parts of a second, repre- 
sented by A£, and let AY be the increment of velocity in 
the time M on this supposition. JSTow f represents the 
increment of velocity in eacli second, and aY the increment 
of velocity in &t seconds: moreover the force P is supposed 
constant d iring At, so that the motion is uniformly accele- 
rated during that time (Art. 44.). 



32 RELATIONS OF 

Now this is true (if the supposition, that P remains constant 
during the time M, on which it is founded, be true), how- 
ever small the time &t may be. But if this time be 
infinitely small, the supposition on which it is founded is in 
all cases true, for P may in all cases be considered to remain 
the same during an infinitely small period of time, although 
it does not remain the same during any time which is not 

aV dV 
infinitely small. !N"ow when At is infinitely small— -=-,- ; 

generally therefore f= -== . 

If Y increase as the time t increases, or if the motion be 
accelerated, then -37 is necessarily a positive quantity. If, 
on the contrary, Y diminishes as the time increases, then 
-57 is negative ; so that, generally, 

f*±m w> 

the sign ± being taken according as the motion is accele- 
rated or retarded. Substituting this value of f in the last 
proportion we have in the case, in which P represents a 
variable pressure, 

*-*?£• » 

The principles stated above constitute the fundamental rela- 
tions of pressure and motion. 



96. The velocity Y at any instant of a body moving with 
a variable motion, being the space which it would describe 
in a second of time, if at that instant its motion were to 
become uniform, it follows, that if we represent by M any 
number of seconds or parts of a second, beginning from that 
instant, and by AS, the space which the booy would describe 

* Note (r) Ed. App. 



PRESSURE AND MOTION. 83 

in the time &t, if its motion continued uniform from the com- 
mencement of that time, then, 

At 

Now this is true if the motion remain uniform during the 
time At, however small that time may be, and therefore if it 
be infinitely small. But if the time At be infinitely small, 
the motion does remain uniform during that time, however 
variable may be the moving force ; also when At is infi- 
nitely small, — = -jr. Therefore, generally, 

*«S a* 

The equations (73) and (74) are the fundamental equations 
of dynamics: they involve those dynamical results which 
have been discussed on other principles in the preceding 
parts of this work.* 



The Descent of a Body upon a Curve. 

*97. If the moving force P upon a body varies directly as its 
distance at any time from a given point towards which it 
falls, then the whole time of the body' s falling to that 
point will be the same, whatever may be the distance from 
whwh it falls. 

Let A be the point from which the body falls, and B a 
point towards which it falls along the path 

\^ APB, which may be either curved or straight ; 

also let the body be acted upon at each 
point P of its path, by a force in the direc- 
tion of its path at that point which varies as 

* Thus if the latter equation be inverted, and multiplied by the former, we 
obtain the equation 

r dt ± g • y \dt> ±2g K dt P 
dS "IW ' 

/.Si 
PrfS, 

VY »» 

which is identical with equation (47). 



84: HI I.ATIOXS OF 

its distance BP, measured along the path from B; the time 
of falling to B will be the same, whatever may be the dis- 
tance of the point A from which the body falls. 

For let BP=S, and let the force impelling the body 
towards B be represented by cS, where c is a constant quan- 
tity ; suppose the body, instead of falling from A towards 
B, to be projected with any velocity from B towards A, and 
let v be the velocity acquired at r, and V that at A, and 
let BA=S 1? then by equation (47), 



v'-^-I/m^-^ 



°9 t 



Suppose now the velocity of projection from B to have 
been such as would only just carry the body to A, so that 

v=o, 

.•.o'=f(S,'-S') (75). 

"Now by equation (74), 

dt_l ._pd£ 

p as 

and if -JT represent the whole time in seconds occupied in 
the ascent of the body from B to A, 

- *®\ 

It is clear that the time required for the body's descent 
from A to B is equal to that necessary for the ascent from 
B to A, so that the whole time required to complete the 
ascent and descent is equal to T, and is represented by the 
formula 

i 

t (76). 



'=© 



PRESSURE AND MOTION. 85 

Now this expression does not contain S 1? i. e. the distance 
from which the body falls to B ; the time T is the same 
therefore, whatever that distance may be. 



The Simple Pendulum. 

98. If a heavy particle P he imagined to he suspended from a 
point Chy a thread without weight, and allowed to oscillate 
freely, hut so as to deviate hut little on either side of the 
vertical, then will its oscillations, so long as they are thus 
small, he performed in the same time whatever their ampli- 
tudes may he. 

For let the inclination PCB of CP to the vertical be repre- 
sented by $, and let the weight w of the particle 
A P, which acts in the direction of the vertical VP, 
| v /] be resolved into two others, one of which is in the 
| / ; direction CP, and the other perpendicular to that 
v \J direction : the former will be wholly counteracted 
^ — ^ by the tension of the thread CP, and the latter will 
be represented by w sin. YFC=w sin. d ; and, act- 
ing in the direction in which the particle P moves, this will 
be the whole impressed moving force upon it (Art. 92.) Now 
so long as the arc d is small, this arc does not differ sensibly 
from its sine, so that for small oscillations the impressed mov- 
ing force upon P is represented by w&, or by— ^=--, or by — , 

if I represent the length CP of the suspending thread, and S 
the length of the arc BP. Now in this expression w and I 
are constant throughout the oscillation, the moving force va- 
ries therefore as S. Hence by the last proposition, the small 
oscillations on either side of CB are isochronous, since so long 
as they are thus small, the impressed moving force in the 
direction of the motion varies as the length of the path BP 
from the lowest point B. Since in the last proposition the 
moving force was assumed equal to dS, and that here it is 

w w 

represented by -i-S, therefore in this case c=—. Substitut- 
o o 

ing this value in equation (76), 

*=$** (77) - 

A single particle thus suspended by a thread without 



M> mi; PABALLELOGRAM OF MOTION. 

weight, is thai which i- meant by a simim.i: i>kndulum. It i& 
evident that the time of oscillation increases witli the length 
I of the pendulum. 

Impulsive Force. 

99. If any number of different moving forces be applied 
to as many equal bodies, the velocities communicated to 
them in the same exceedingly small interval of time, will be 
to one another as the moving forces. For let P„ P 2 , repre- 
>ent the moving forces, and f, f, the additional velocities 
they would communicate per second if each moving force 
remained continually of the same magnitude (Art. 93.), then 
would tf, tf, be the whole velocities communicated on this 
supposition in t seconds ; let these be represented by Y„ Y 2 ; 
therefore by Art. 94. 

P^P,::/, :/,::*/, :^::Y,:Y, 

The proposition is therefore true on the supposition that Y x 
and r 2 remain constant during the interval of time t ; but 
if t be exceedingly small, then whatever the pressures P, 
and P 2 may be, they may be considered to remain the same 
during that time, therefore the proposition is true generally, 
when, as above, the moving forces are supposed to act on 
equal bodies, or successively on the same body, through 
equal exceedingly small intervals of time. 

Moving forces thus acting through exceedingly small in- 
tervals of time only, are called impulsive forces. 



The Parallelogram of Motion. 

100. If two impidsive forces P„ P 2 , whose directions are AB 

and AC. he impressed at the same time upon 

J~~^7* a body at A, which if made to act upon it 

L^^ / separately would cause it to move through 

AB and AC in the same given time, then 

will the body be made, by the simultaneous action of these 

impulsive forces, to describe in that time the diagonal AD 

of the parallelogram, of which AB and AC are adjacent 

sides. 

For the moving forces P 2 and P a acting separately upon 



INDEPENDENCE OF SIMULTANEOUS MOTIONS. ST 

the same body through equal infinitely small times, com oiu- 
nicate to it velocities which are (Art. 99.) as those foices ; 
therefore the spaces AB and AC described with these velo- 
cities in any given time are also as those forces. Since then 
AB and AC are to one another as the pressures P, and P 2 , 
therefore by the principle (Art. 2.) of the parallelogram of 
pressures, the resultant R of J > 1 and P 2 is in the direction of 
the diagonal AD, and bears the same proportion to P x and 
P 2 that AD does to AB and AC. 

Therefore the velocity which the resultant E of P, and P 2 
would communicate to the body in any exceedingly small 
time is to the velocities which P 1 and P 2 would separately 
communicate to it in the same time as AD to AB and AC 
(Art. 99.), and therefore the spaces which the body would 
describe uniformly with these three velocities in any equal 
times are in the ratio of these three lines. But AB and AC 
are the spaces actually described in the equal times by rea- 
son of the impulses of P x and P 2 . Therefore AD is the space 
described in that time by reason of the impulse of R, that is, 
by reason of the simultaneous impulses of P x and P 3 . 



101. THE INDEPENDENCE OF SIMULTANEOUS MOTIONS. 

It is evident that if the body starting from A had been 
made to describe AB in a given time, and then 
/*~^==^ D had been made in an equal time to describe 
^^Hl / BD, it would have arrived precisely at the same 
point D to which the simultaneous motions 
AC and AB have brought it, so that the body is made to 
move by these simultaneous motions precisely to the same 
point to which it would have been brought by those motions, 
communicated to it successively, but in half the time. The 
following may be taken as an illustration of this principle of 
the independence of simultaneous motions. Let a canal-boat 
be imagined to extend across the whole 
width of the canal, and let it be supposed 
that a person standing on the one bank at 
A is desirous to pass to a point I) on the 
opposite bank, and that for this purpose, as the bout passes 
him, he steps into it, and walks across it in the direction 
AB, arriving at the point B in the boat precisely at the in- 
stant when the motion of the boat has carried it through 
BD ; it is clear that he will be brought, by the joint effect 





B8 THE POLYGON OF MOTIOK. 

of his own motion across the boat and the ooat y s motion 
along the canal, to the point I) (haying in reality described 
the diagonal A IK which point he wonld have reached in 
doable tne time if he had walked across a bridge from A tc 
B in the Bame time that it took him to walk across the boat, 
and had thru in an equal time walked from B to D along 
the opposite Bide. 



The Polygon of Motion. 

102. Let any number of impulses be communicated simul- 
taneously to a body at O, one of which 
would cause it to move from A to O in a 
given time, another from B to O in the 
same time, a third from C to O in that time, 
and a fourth from D to O. Complete the 
parallelogram of which AO and BO are ad- 
jacent sides ; then the impulses AO and BO would simulta- 
neously cause the body to move from E to O through the 
diagonal EO in the time spoken of. Complete the parallelo- 
gram E( >CF, and draw its diagonal OF, then would the im- 
pulses EO and CO, acting simultaneously, cause the body to 
move through FO in the given time : but the impulse EO 
produces the same effect on the body as the impulses A( ) 
and BO; therefore the impulses AO, BO, and CO, will 
her cause the body to move through FO in the given 
time. In the same manner it may be shown that the im- 
pulses AO, BO, CO, and DO, will together cause the body to 
move through GO in a time equal to that occupied by the 
body's motion through any one of these lines. 

It will be observed that GD is the side which completes 
the polygon OAEFG, whose other sides OA, AE, EF, FG, 
are respectively equal and parallel to the directions OA, OB, 
< X '. and OD, of the simultaneous impulses. 

[instead of the impulses AO, 6cc. taking place simultane- 
ously, if they had been received successively, the body 
moving first from O to A in a given time; then through 
AE. which is equal and parallel to OB, in an equal time; 
then through EF, which is equal and parallel to OC, in that 
time : and lastly through FG, which is equal and parallel to 
OD, in that time, it would have arrived at the same point G, 
to which these impulses have brought it simultaneously, but 
after a period as many times greater as there are motions, so 



89 

that the principle of the independence of simultaneous 
motions obtains, however great may be the number of such 
motions. 

The Principle of D'Alembert. 

103. Let "W,, "W a , Tv~ 3 , &c. represent the weights of any 
number of bodies in motion, and P 1? P 2 , P 3 , &c. the moving 
forces (Art. 92.) upon these bodies at any given instant of 
the motion, i. e. the unbalanced pressures, or the pressures 
which are wholly employed in producing their motion, and 
pressures equal to which, applied in opposite directions, 
would bring them to rest, or to a state of uniform motion. 

Then (Art. 95.), P.= y/„ 7 *=^f» P, = ^/„&c 

where f,f,f-, &"c. represent the additions of velocity which 
the bodies would receive in each second of time, if the 
moving force upon each were to become, at the instant at 
which it is measured, an uniform moving force. Suppose 
these bodies, whose weights are W l5 "W 2 , W 3 , &c. to form a 
system of bodies united together by any conceivable mecha- 
nical connection, on which system are impressed, in any 
way, certain forces, whence result the unbalanced pressures 
P x , P 2 , P 3 , &c. on the moving points of the system. Now 
conceive that to these moving points of the system there are 
applied pressures respectively equal to P 1? P 2 , P 3 , &c. but 
each in a direction opposite to that in which the motion of 
the corresponding point is accelerated or retarded. Then 
will the motion of each particular point evidently pass into 
a state of uniform motion, or of rest (Art. 92.). The whole 
system of bodies being thus then in a state of uniform 
motion, or of rest, the forces applied to its different elements 
must be forces in equilibrium. 

"Whatever, therefore, were the forces originally impressed 
upon the system, and causing its motion, they must, together 
with the pressures P l5 P„, P 3 , cv/c. thus applied, produce a 
state of equilibrium in the system; so that these forces 
ginally impressed upon the system, and known in Dynamics 
as the impressed forces) have to the forces P„ P,. P , &C, 
when applied in directions opposite to the motions of their 
several points of application, the relation of forces in equili- 
brium. The forces P„ P 2 , P„ <fcc. are known in Dynamics 
as the effective forces. Thus m amy system of bodies 
mechanically connected in any way, so that their motions 



Tin; pkinltpi B *9 d'alembeet. 

mutually infiuen mother, if forces equal to tht 

■ /, applied in directions opposite to their 
actual directions, tins, would he in equilwrium with the 
impressed forces, which is the principle of D'Alembert. 



104. Tht work accumulated in a moving body through any 
gpod is equal to the work which must he done upon it. in 
(in i>i>j>^s',t, direction, to overcome the effective force upon 
it through that space. 

This is evident from Arts. P>S. and fif).. since the effective 
force is the unbalanced pressure upon the body. 

If the work of the effective force be said to be done upon 
the body,* then the work of the effective force upon it is 
equal to the work or power accumulated in it. and this work 
of the effective force may be all said to be actually accu- 
mulated in the body as in a reservoir. 



Motion of Translation. 

Definition. — AVhen a body moves forward in space, with- 
out at the same time revolving, so that all its parts move 
with the same velocity and in parallel directions, it is said to 
move with a motion of translation only. 



105. In order that a body may move with a motion of trans- 
lation o?dy. the resultant of the forces impressed upon it 
must have Us direction through the centre of gravity of 
the body. 

For let ?/',, i0„ u\. &c. represent the weights of the parts 
or elements of the Kdy, and let f represent the additional 
velocity per second, which any element receives or would 
receive if its motion were at any instant to become uniformly 
accelerated. Since the motion is one of translation only, 
the value of f is evidently the same in respect to every 
r.ther element. The effective forces P„ P„, P s , &c. on the 
different elements of the body are therefore represented bv 
www 
— /, — /,— f, t*:e. &c. 

g J g J g m 

* This cannot perhaps be correctly said, since work supposes resistance. 



MOTION OF ROTATION. 91 

ISTow the forces P 1? P 2 , P„ &c. are evidently parallel pres- 
sures. Let X be the distance of the centre (see Art. IT.) of 
these parallel pressures from any given plane ; and let a?,, aj„ 
a? 3 , &c. be the perpendicular distances of the elements w„ w» 
w 3 , &c. that is, of the points of application of P n P 2 , P 8 , &c. 
from the same plane. Therefore (by equation 18), 

{P 1 + P 3 + P 3 + { X=P 1 « 1 +P a a>,+P 1 ®,+ . . . . ; 

,j ?/+?/+?/+ ....|x= ?A+>+5A+ ..., 

/. {^4-^4-^3+ • • •} X=w 1 a5 1 -fw a oj a +w,a?,+ . . ., 
^ w 1 x 1 -\-w 2 %s + w z % 3 + .... 



W 1 +W a + W,+ .... 

But this is the expression (Art. 19.) for the distance of the 
centre of gravity from the given plane ; and this being true 
of any plane, it follows that the centre of the parallel pres- 
sures P l5 P 2 , P 3 , &c. which are the effective forces of the 
system, coincides with the centre of gravity of the system, 
and therefore that the resultant of the effective forces passes 
through the centre of gravity. Now the resultant of the 
effective pressures must coincide in direction with the result- 
ant of the impressed pressures, since the effective pressures 
when applied in an opposite direction are in equilibrium 
with the impressed pressures (by D'Alembert's principle). 
The resultant of the impressed pressures must therefore have 
its direction through the centre of gravity. Therefore, &c. 



Motion of Potation about a fixed Axis. 

106. Let a rigid body or system be capable of motion 
about the axis A. Let m n m 2 , m 3 , &c. represent the volumes 
of elements of this body, and /x the weight of each unit 
of volume. Also let f x ,f»f^ &c represent the increments 
of velocity per second, communicated to these elements 
respectively by the action of the forces impressed upon the 
system. Let P n P 2 , P 3 , &c. represent these impressed forces, 
smdp^p^ &c. the perpendicular distances from the axis at 
which they are respectively applied. 

Now since fj-m,, i*m„ M-m 3 , &c. are the weights of the ele- 
ments, smdf^f^ &c. the increments of velocity they receive 



MOTION OF ROTATION. 

Becond, it follows that^/, ^/ 2 , !^/„ &c are 
9 9 9 

the effective forces apoD them (Art. L03.). Let p,, p f , p 3 . &c 
represent the distances of these elements respectively from 
the axis of revolution, then Bince their effective forces are 
in directions perpendicular to these distances, the moments 

of these effective forces about the axis are \—ifo. [ -Li fa 

9 9 

^^/ 3 p 3 , etc. Also Pj? I? Pj? 2 , P 3 j9 3 , &c. are the moments of 

9 
the impressed forces of the system about the axis. Now the 
impressed forces P„ P 2 , P 3 , &c, together with the resistance 
of the axis, which is indeed one of the impressed forces, are 
in equilibrium with the effective forces by D'Alembert's 
principle. Taking then the axis as the point from which the 
moments are measured, the sum of the moments of P„ P„ 
&c must equal the sum of the moments of the effective 
forces, or 

9 ^ 9 

Now let/ represent that value of f^f^ etc. which corres- 
ponds to a distance unity from the axis. Since the system 
is rigid, and f, f x , f^ tfce. represent arcs described about 
it in the same time at the different distances 1, p„ p 2 , etc. it 
follows that these arcs are as, their distances, and therefore 
Jihat/ l =/p I ,/,=/p a ,/ i ==/p t , &c. Substituting these values 
in the preceding equation, we have 

g & 



/-Kr 


;+mj;+ . . . 


.}=P 


i>i+I> a 


+ 




or f-'S.mf- 


=2Pp. 


• • > 






. f _9 *Pp 

M- I 




• (78), 





where I represents the moment of inertia of the mass about 
its axi> of revolution.* 

* If a represent the angular velocity, or the velocity of an element at dis- 
tance unity, then by equation (7 2), / = -f- -^, . • . a-^= -f- -^ SPpa ; 



MOTTOX OF EOTATIOX. 



93 



107. If the impressed forces P be the weights of the parts 
of the "body and 6 be, in any position of 
_^ the body, the inclination to the vertical 
Ay of the line AG, drawn from A to the 
centre of gravity G, then since the sum of 
the moments of the weights of the parts is 
equal to the moment of the weight of the 
whole mass collected in its centre of 
gravity (Art. 17.), we have, representing 




AG by G, 

2Pp=Mj* . GG;=]n> . G. sin. 

therefore ^equation 78),/": 



MG . 
g-j- sm. 



(79). 



108. To find the resultant of the effective forces on a body 
which revolves about a fixed axis. 

The resultant of the effective forces upon a body which 
revolves about a fixed axis, is evidently equal to that single 
force which would just be in equilibrium with these if there 
were no resistance of the axis. Let K be that single force, 
then the moment of K about any point must equal the sum 
of the moments of the effective forces about that point. 

Take a point in the axis for the point 
about which the moments are measured, 
and let L be the perpendicular distance 
from A of the resultant E. ]STow, as in 
Art. 106. it appears that the sum of the 
moments of the effective forces about A is 

represented by /-^rap 3 , 




11 
Now pa is the velocity of a point at distance p t therefore Ypu is the work 

(Art. 50.) of the force P per second ; therefore / Ypadt is the work of P 


(equation 40) in the time t, which is represented by U, therefore oi" — a.," 1 

— -\ — '-j- which corresponds with the result already obtained. See equation 

(61). M 



94 MOTION OF ROTATION. 



.-. KL=/^V (80). 

To determine the value of R let it be observed that tlic 
effective force -f / m 1 p 1 on any particle m x , acting in a direc- 
tion n x m x , perpendicular to the distance Am 1 from the axis 
A, may be resolved into two others, parallel to the two 
rectangular axes Ay and Ax, each of which is equal to the 
product of this effective force, whose direction is n x m x , and 
the cosine of the inclination of n l m 1 to the corresponding 
axis. Now the inclination of m x ii x to Ax is the same as the 
inclination of Am 1 to Ay, since these two last lines are per- 
pendicular to the two former. The cosine of this inclination 

AN v 

equals therefore 1 or *±, if AN 1 =y l . Similarly the cosine 

Am x p x 

of the inclination of n x m x to Ay equals 1 or -i , if AM X = x x . 

Am, a 

The resolved parts in the directions of Ax and Ay of the 
effective force - fm x p x are therefore - fmj x ^, and - fm x p x 

g g pi g 

% or t fm x y x and t fm x x x . 

Similarly the resolved parts in the directions of Ax and 
Ay of the effective force upon m 2 are -fmj/s and - fm^x^ 

g g 

and so of the rest. 

The sums X and Y of the resolved forces in the directions 
of Ax and Ay respectively (Art. 11.) are therefore 

g g g 

(m&-fm x x x + ^fm i x a + ^fmsX 3 + . . . =Y; 

g g g 

°r^f{m x y x + m$, + m a y i + }=X, 

9 

and - f \m x x x + m 2 a? 2 + m % x % + } = Y. 

9 

Now let G x and G 2 represent the distances G 2 G and Gt x G 
of the centre of gravity of the body from Ay and Ax respec- 
tively, and let the whole volume of the body be represented 
byM, 



MOTION OF ROTATION. 95 

A (equation 18), MG 2 =m I y 1 + m s y,+?^ a + . . . ., 
MG X — m 1 cc 1 +m 2 a? 2 + m 3 a? 3 + . . . .; 

/.X=^/MG 2 , Y=J/M€fc (81). 



fcw,<v ^ l^Tow (Art. 11.), E= VX' + Y 2 , therefore 

! n\) Now if G be the distance AG of the 

A centre of gravity from A, G = VG^ + G/, 

.\K=£/MG (82). 

Substituting in equation (82) the value off from equation 
(78,) we have 

E = ^££ (83) . 

And substituting in equation (80) for R its value from 
equation (82), 

J g J g 9 

where L is the distance of the point of application of the 
resultant of the effective forces from the axis. 

Now let 6 be the inclination of the resultant R to the 
axis Ax, 

.\ (Art. 11.), R cos. a=X, R sin. 4=Y, 

Y 

/.tan. d= y ; but by equations (81), 

Y a AG, 4 ^„ 

/.tan. d=tan. AGG t , :J=AQQ l . 

The inclination of the resultant R to Ax is therefore 
equal to the angle AGG„ but the perpendicular to AG is 
evidently inclined to Ax at this same angle. Therefore the 
direction of the resultant R is perpendicular to the line Ad, 
drawn from the axis to the centre of gravity. Moreover 



Tli: 

:* its point of application 
A h&\ inationa 

The Centre of Pee sg 

at, that if at a point of the body through 
which th< ' vnt of the effective forces ; 

there * r> it? motion, then there will 

he produced upon that obstacle the same effect as though 
the whole of the effective forces were collected in that 
\ and made to act there upon the obsl . — that the 
whole of these forces will take effect upon the obsracle. and 
there will be no effect of these forces produced else- 
where, and therefore no repercussion upon the axis. 
It is tV.r this reason that the point in the resultant, 
where it cuts the line AG drawn from the axis to the 
centre of gravity, is called the centre of percus- 
Its distance L from A is determined by the equation 



G 



- 



which is obtained from equation (84) by writing aTIy 5 for I 
Art v ". . K being the radius of gyration. If at the centre 
of percussion the body receive an impulse when at rest, 
then since the resultant of the effective forces thereby pro- 
duced will have its direction through the point where the 
impulse i's communicated, it follows that the whole impulse 
will take effect in the production of those effective forces, 
and no portion be expended on the axis. 



The Centre of Oscillation. 

11 o. It has been shown 3. | that in the simple pen- 

dulum, supposed to be a single exceedingly small element 
of n. spended by a thread without weight, the time 

- dependent upon the length of this 
thread, or th< ■■• of the suspended element from the 

about which it oscillates. It* therefore we imagine a 
number .-t* BUch elemi be thus suspended ar 

-. and if we suppose them, after 

having been at first united into a continuous body, placed 

>n, all to be released at once from this 



THE CENTRE OF OSCILLATION. 97 

union with one another, and allowed to oscillate freely, it is 
manifest that their oscillations will all be performed in 
different times. ]STow let all these elements again be con- 
ceived nnited in one oscillating mass. All being then com- 
pelled to perform these oscillations in the same time, whilst 
all tend to perform them in different times, the motions of 
some are manifestly retarded by their connexion with the 
rest, and those of others accelerated, the former being those 
which lie near to the axis, and the others those more remote ; 
so that oetween the, two there must be some point in the 
body where the elements cease to be retarded and begin to 
be accelerated, and where therefore they are neither accele- 
rated nor retarded by their connexion with the rest ; an ele- 
ment there performing its oscillations precisely in the same 
time as it wonld do, if it were not connected with the rest, 
bnt suspended freely from the axis by a thread without 
weight. This point in the body, at the distance of which 
from the axis a single particle, suspended freely, would per- 
form its oscillations precisely in the same time that the body 
does, is called the centre of oscillation. 



The centre of oscillation coincides with the centre of 
percussion. 

111. For (by equation 79) the increment of angular velo- 
city per second f of a body revolving about an hori- 
zontal axis, the forces impressed upon it being the 
weights of its parts only, is represented by the for- 

mula g-^-sm. d, where 6 is the inclination to tne ver- 
tical of the line AG, drawn from the axis to its 
centre of gravity. But (by equation 84), L=^p, where L 

is the distance AO of the centre of percussion from the 
axis, 

:.f=^ 6 (86) 

/. fL=g sin. & 

Now it has been shown (Art, 98.), that the impressed 
moving force on a particle whose weight is w, suspended 
from a thread without weight, inclined to the vertical at an 
angle 6, is represented by w sin. 6 ; moreover it'/" represent 

7 



9S 'Hi: i I.A1I0N. 

the increment of velocity per second on this particle, then 
—f is the effective force upon it. Therefore by D'Alem- 

bert's principle, 

w 
wem.»=-f, .: f= ff sin. 6, .: /=/L. 

Now,/L is the increment of velocity at the centre of 
percussion, and/' is that upon a single particle suspended 
freely at any distance from the axis. If such a particle 
were therefore suspended at a distance from the axis equal 
to that of the centre of percussion, 6ince it would receive, 
at the same distance from the axis, the same increments of 
velocity per second that the centre of percussion does, it 
would manifestly move exactly as that point does, and per- 
form its oscillations in the same time that the body does. 
Therefore, &c. 



112. The centres of suspension and oscillation are reci- 
procal. 

Let O represent the centre of oscillation of a body 
when suspended from the axis A ; also let G be its 
centre of gravity. Let AO=L, AG=G, OG=G 1 ; 
also let the radius of gyration about A be repre- 
sented by K 2 , and that about G by k\ Therefore 
(equation 59), K 2 =G 2 + & 2 ; 

G 2 4- h* k* 
;. (equation 85), L= -^=G+ ^ (87), 

.-.G+G^G+J, 

••^=5 (88). 

Now let the body be suspended from O instead of A ; 
when thus suspended it will have, as before, a centre of 
oscillation. Let the distance of this centre of oscillation 
from O be L„ 

jf 
:. by equation (87), L 1 =G 1 + tt-, 



PROJECTILES. 99 

.". by equation (88), 1^=^-1-6=1. 

Since then the centre of oscillation in this second case is at 
the distance L from O, it is in A; what was before the 
centre of suspension has now therefore become the centre 
of oscillation. Thus when the centre of oscillation is con- 
verted into the centre of suspension, the centre of suspen- 
sion is thereby converted into the centre of oscillation. 
This iswhat is meant, when it is said that the centres of 
oscillation and suspension are reciprocal. 



Projectiles. 

113. To determine the path of a body projected obliquely 
in vacuo. 

Suppose the whole time, T seconds, of the flight of the 

body to any given point P 
-'"T of its path, to be divided 
into equal exceedingly small 
intervals, represented by 
aT, and conceive the whole 
effect of gravity upon the 
projectile during each one 
of these intervals to be col- 
lected into a single impulse at the termination of that inter- 
val, so that there may be communicated to it at once, by 
that single impulse, all the additional velocity which is in 
reality communicated to it by gravity at the different periods 
of the small time aT. 

Let AB be the space which the projectile would describe, 
with its velocity of projection alone, in the first interval of 
time ; then will it be projected from B at the commence- 
ment of the second interval of time in the direction ABT 
with a velocity which would alone carry it through the dis- 
tance BK=AB in that interval of time ; whilst at the same 
time it receives from the impulse of gravity a velocity such 
as would alone carry it vertically through a space in that in- 
terval of time which may be represented by BF. 1 \y reason 
of these two impulses communicated together, the body will 
therefore describe in the second interval of time the diago- 
nal BC of the parallelogram of which BE and BF are adja- 

LOFC. 




100 PROJECTILES. 

cent sides. At the commencement of the third interval ii 
will therefore bave arrived at 0, and will be projected from 
thence in the direction BCX, with a velocity which would 
alone cany it through CX=BC in the third interval ; whilst 
at the same time it receives an impulse from gravity com- 
municating to it a velocity which would alone carry it 
through a distance represented by CG=BF in that interval 
of time. These two impulses together communicate there- 
fore to it a velocity which carries it through CD in the third 
interval, and thus it is made to describe all the sides of the 
polygon ABCD ... P in succession. Draw the vertical PT, 
and produce AB, BC, CD, &c. to meet it in T, N, O . . ., 
and produce GC, HD, &c. to meet BT in K, L, &c. 

Now, since BC is equal to CX, and CK is parallel to XL, 
therefore KL is equal to BK or to AB. 

Again, since CD is equal to DZ, and DL is parallel to ZM, 
therefore LM is equal to KL or to AB ; and so of the rest. 

If therefore there be n intervals of time equal to aT, so 
that there are n sides AB, BC, CD, <fcc. of the polygon, and 
n divisions AB, BK, &c. of the line AT, then AT,=nAB and 
BT=(#»— 1)AB, 

/.TN=(t*-1)KC=(ti-1)BF. 

Similarly CN=(?i-2)CX, therefore NO = (ti-2)DX= 
(n— 2)BF ; and so of the remaining parts of TP. 

Now these parts of TP are (n—1) in number, therefore 
TP=(n-l)W+(n-2)W + (?i-3)W+ ... {(ra-1) terms}; 
orTP={(n— l)+(n-2)+ . . . (BF. 

Therefore, summing the series to (n—1) terms. 
TP=f2(»-l)-(»-8)}p=!) . BF, 

a 

Now g represents the additional velocity which gravity 
would communicate to the projectile in each second, if it 
acrid upon it alone. g&T is therefore the velocity which it 
would communicate to it in each interval of aT seconds. 
g^T is therefore the velocity communicated to the body by 
each of the impulses which it has been supposed to receive 
from gravity. 



PROJECTILES. 101 

Now BF is the space through which it would be carried 
in the time aT by this velocity, 



Also aT=- 



•.BF=(^T)aT=<7(aT) 3 , 



/.TP=i^-l)J=^(l- l)r. 

Now this is true, however small may be the intervals of 
time aT, and therefore if they be infinitely small, that is, if 
the impulses of gravity be supposed to follow one another at 
infinitely small intervals, or if gravity be supposed to act, as 
it really does, continuously. 

But if the intervals of time aT be infinitely small, then 
the number n of these intervals which make up the whole 
finite time T, must be infinitely great. Also when n is infi- 
nitely great, -=0. 
n 

In the actual case, therefore, of a projectile continually 
deflected by gravity, the vertical distance TP between the 
tangent to its path at the point of projection, and its position 
P after the flight has continued T seconds, is represented by 
the formula 

TP=igT (89). 

Moreover AT=tiAB, and AB is the space which the body 
would describe uniformly with the velocity of projection in 
the time aT, so that nAB is the space which it would de- 
scribe in the time n . aT or T with that velocity. If there- 
fore Y equal the velocity of projection, then 

AT=Y . T . . . . (90) ; 

so that the position of the body after the time T is the same 
as though it had moved through that time with the velocity 
of its projection alone, describing AT, and had then fallen 
through the same time by the force of gravity alone, describ- 
ing TP (see Art. 101.). 



A 



114. Let AM=#, MP=y y angle of 
projection TAM=a, velocity of projec- 
tion =V. 



\ •: 



=1 T..-.T= 

xtaiL«-v=MT-Ml > =TP=i : 7r 91). 

-ututing the vain :om the preceding equation, 

— .7 * — . . — 

_ 

H be the height through which a bodv must fall fit 
gravity 1 ~_e height d 

that V*=2 B ..." - '.-'....:.. -IL = ~ 

~- - - 

y=~ -^, (92). 



115. To find the time cf tike fight qf a project 

It has been shown (equation 91), that if T represent the 
time in seconds of the fGght to a point whose co-ordinates 
are x and y. then 

.-, 
'= .-:.:. — .. .-.P=--jrtam a— 

~ 

•'• T = \ : . _ (93). 



~ — ~ -= — near".---. :.T=iJxtKML. m — % n earl v. 
4 16 " x 

If the projectile descend again to the horizontal plane from 
which it was projected. andT be the whole tim 
and X .:mge upon the plane, then, since at tL 

piration of the tin. J. = =X. 



.'. T= \ V X = r s Iran. « nearly. 



PROJECTILES. 



10c 



116. To find the greatest horizontal distanced, to which a 
'projectile ranges, having given the elevation a and the 
velocity Y of its projection. 

When the projectile attains its greatest horizontal range, 
its height y above the horizontal plane 
becomes 0, whilst the abscissa x of the 
point P, which it has then reached in 
its path, becomes X. Substituting 
these values and X, for y and x in 
equation (92), we have 0=X tan. a— 
X 2 sec. 2 a 




4H ' 

.*. X=4H tan. a cos. 3 a=4H sin. a cos. a. 

.\X=2Hsin. 2a (94). 

If the bod j be projected at different angular elevations, 
but with the same velocity, the horizontal range will be the 

IT 



greatest when sin. 2a is the greatest, or when 2a =-, or a 



2' 



117. To find the greatest height which a projectile will 
attain in its flight if projected with a given velocity r , and 
at a given inclination to the horizon. 




Multiplying both sides of equation 

(92) by 4H cos. 2 a, we have 4H cos. 8 a 

. y=4H cos. 2 a tan. a . x—x i =2H. (2 

cos. a sin. a) x— a? 2 =2H sin. 2a . x—x\ 

^ Subtracting both sides of this equa- 

^ \ tion from H 2 sin. 2 2a, we have 

H 2 sin 2 2a-4H cos. 2 a . y=H 2 sin. 2 2a-2H sin. 2a . x + x\ 

But sin. 2 2a=4 sin. 2 a cos. 2 a, 

.\4H cos. 2 a JH sin. 2 a -yl z= {II sin. 2a— &>}*. . . . (95). 

Now the second member of this equation is always a 
positive quantity, being a square. The first member Is 
therefore always positive; therefore TI sin. 2 a— y is always 
positive. Whence it follows that y can Dever exceed II 
sin. 2 a, so that it attains its greatest possible value when it 
equals II sin. 2 a, a value which it manifestly attains when 



tlie first member of the above equation vanishes, or when 
; -i. 2a, that is, whan becomes equal to half the 

horizontal nu _ 'parent from the last pro- 

: hat the greatest height BD (A' the projectile 
presented by 11 Bin." a, a height which it attains when 
A 1 » equals halt* the horizontal range. 



118. The path of a projectile in vacuo is a parabola. 

Let B be the highest point in the 
flight of the projectile, and BD its 
greatest height. Draw PMj perpen- 
dicular to BD. Let BM,=x l3 M,P 

A * ^ =*■ 

.-. a^BD— MJD=BD— PM=H sin.' a— y , 

y l= DM=AM— AD=a?— H sin. 2a. 

Substituting these values in equation (95), 

^■=±11 cos. 2 tt . x x (96), 

which is the equation to a porabola whose vertex is in 
P>. whose axis coincides with BD, and whose parameter is 
■ill cos. 3 a. 

The path of a projectile in vacuo is therefore a parabola, 
whose vertex is at the highest point attained by the pro- 
jectile, and whose axis is vertical. 



119. To find the range of a projectile upon an inclined 

plane. 

Let B represent the range AP of a projectile upon an 

B inclined plane AB. whose inclination is 

/^Z-^TX '• Th en H and a. being taken to repre- 

±/<^^ Ben * r ^ ie same quantities as before, and 

M , y being the co-ordinates of P to the 

horizontal azis A< . we have " 

»=AM=AP cos. PAM=K cos. 1, 
y=PM=AP sin. PAM=B sin. «. 

^ Substituting these values of x and y in the general equa- 
tion (92) of the projectile we have 



R sin. i=R cos. » tan. a— 



PROJECTILES. 105 

R 2 cos. 2 i sec. 2 a 



4H 

Dividing by R, multiplying by cos. a, and transposing 

R cos. 3 1 sec. a , m 

jtj =cos. i sm. a — sin. » cos. a=sm. (a — i), 

... E=4 H- 8 - ^-) cos - ■ (97) . 

COS. I J 

ISTow sin. (2a — i) — sin. »=sin. {a + (a — »)| — sin. {a — (a— 
»)} =2 sin. (a — ») cos. a. 

Substituting this value of 2 sin. (a — ») cos. a in the pre- 
ceding equation, we have 

R=2H , S in.( 2r .) sin.n 



Now it is evident that if a be made to vary, < remaining 
the same, R will attain its greatest value when sin. (2a — ») 
is greatest, that is when it equals unity, or when 2a — »= 
it if i 

~, or when a=--f-- This, then, is the angle of elevation 

corresponding to the greatest range, with a given velocity 
upon an inclined plane whose inclination is ». 

If in the preceding expression for the range we substitute 

) o" - ( a— ') t f° r °S the value of the expression will be found 

to remain the same as it was before ; for sin. (2a— i) will, by 
this substitution, become sin. {*— 2(a— i)— i\ =sin. j^— (2a 
— »)J =sin. (2a— »). The value of R remains therefore the 

if 
same, whether the angle of elevation be a or ^— -(a— «). 

And the projectile will range the same distance on the 
plane, whether it be projected at one of these angles of 
elevation or the other. 

Let BAC be the inclination of the plane on which the 
projectile ranges, and AT the direc- 
1 ' //\ 4 tion of projection. Take DAS equal 

// X to BAT. Then BAT=TAC-BAO 

// \ = a -i. And SAC=DAC-DAS= 

xJ^^^l:;~:, « --BAT= -(«-.). TherangeAP 
is therefore the same, whether TAC or SAC be the angle of 



106 CD NTBIFUGAL FORCE. 

elevation, and therefore whether AT or AS be the direction 
of projection. 

Draw AE bisecting the angle BAD, then the angle EAC 

=BAO+BAE=BAO+iBAD=i+J^-i)=j+^ 

The angle EAC is therefore that corresponding to the 
greatest range, and AE is the direction in which a body 
should be projected to range the greatest distance on the 
inclined plane AB. 

It is evident that the directions of projection AS and AT, 
which correspond to equal ranges, are equally inclined to 
the direction AE corresponding to the greatest range. 



120. The velocity of a projectile at different points of its 
path. It has been shown (Art, 56.), that if a body move in 
any curve acted upon by gravity, the work accumulated or 
lost is the same as would be accumulated or lost, provided 
the body, instead of moving in a curve, had moved in the 
direction of gravity through a space equal to the vertical 
projection of its curvilinear path. 

Thus a projectile moving from A to P will accumulate or 
lose ;i quantity of work, which is equal to that which it would 
accumulate or lose, had it moved vertically from M to P, or 
from P t<» M 3 PM bein<* the projection of its path on the 
direction of gravity. Now the work thus accumulated or 
lost equals one half the difference between the vires vivce at 
the commencement and termination of the motion. 

Let Y equal the velocity at A, and v equal the velocity at 

TV TV 

P, therefore the work =% — Y a — i — v*. Moreover, the work 

«/ y 

TV TV 

done through- PM=W . PM, therefore £— Y 3 -£ v<= 

9 9 

TV . PM, therefore Y 2 -V=^MP. Let PM=y, 

;.v>=Y*-2gy (99), 

which determines the velocity at any point of the curve. 



Centrifugal Force. 
121. Let a body of small dimensions move in any curvi- 




CENTRIFUGAL FORCE. 107 

linear path AB, impelled continaally towards 
a given point S (called a centre of force) by a 
given force, whose amount, when the body 
has reached the point P in its path, is repre- 
sented by F-* Let PQ be an exceedingly 
small portion of the path of the body, and 
conceive the force F to remain constant and 
parallel to itself, whilst this portion of its path is being de- 
scribed. Then, if PR be a tangent at P, and QR be drawn 
parallel to SP, PR is the space which the body would have 
traversed in the time of describing PQ, if it had moved 
with its velocity of projection from P alone, and had not been 
attracted towards S, and RQ or PT (QT being drawn paral- 
lel to RP) is the space through which it would have fallen 
by its attraction towards S alone, or if it had not been pro- 
jected at all from P.f Let v represent the velocity which 
it would have acquired on this last supposition, when it 
reached the point T. ' Therefore (Art. 66.), if w represent the 
weight of the body, 

FxPT=i%\ 
9 

Now the velocity v, which the body would have acquired in 
falling through the distance PT by the action of the constant 
iorce F, is equal to double that which would cause it to de 
scribe the same distance uniformly in the same time.J 

Representing therefore by Y the actual velocity of the 
body in its path at P, we have 

iv PT oT7 PT 

Substituting this value of v in the preceding equation, 

* The force here spoken of, and represented by F, is the moving force, or 
pressure on the body (see Art. 92.), and is therefore equal to that pressure 
which would just sustain its attraction towards S. 

f See Art. 113. (equations 89 and 90) ; what is proved there of a body acted 
upon by the force of gravity which is constant, and whose direction La con- 
stantly parallel to itself, is evidently true of any other constant force similarly 
retaining a direction parallel to itself. To apply the same demonstration to 
any such case, we have only indeed to assume g to represent another number 
than 32f 

\ If /represent the additional velocity per second which F would com- 
municate to the body, and t the time of describing PT, then (Art 1 1.) 

v=ft; but (Art. 46.) YT=tfP = (t)t=~t ; so that | is the velocity witb 

which PT would be described uniformly iu the time t. 



108 .:. FORCE. 



Fxi'T=.'"\(!.' T .V, .-.F=2^V QE - 



"V( r, T 



(PR)' 

Now let a circle PQY be described having a common tan 
gent with the curve AJB in the point I\ and passing through 
the point Q. Produce PS to intersect the circumference of 
this circle in V, and join QV ; then are the triangles PQV 
and QPK similar, for the angle RQP is equal to the angle 
QPY (QE and VP being parallel), and the angle QPR is 
equal to the angle QVP in the alternate segment of the cir- 
cle. Therefore p7-v = pv> therefore QR^^—-. Substi- 
tuting this value of QR in the last equation, we have 

gFY'XPBJ' 

Now this is true, however much PQ may be diminished. 
Let it be injinitdij diminished, the supposed constant amount 
and parallel direction of F will then coincide with the actual 
case of a variable amount and inclination of that force, the 

PQ 

ratio — | will become a ratio of equality, and the circle 

PQY will become the circle of curvature at P, and PY that 
chord of the circle of curvature, which being drawn from P 
passes through S. Let this chord of the circle of curvature 
be represented by C, 

■•■^-gO (1 ° 0) - 

The force or pressure F thus determined is manifestly 
exactly equal to that force by which the body tends in its 
motion continually to fly from the centre S, and may there- 
fore be called its centrifugal force. This term is, however, 
generally limited in its application to the case of a body re- 
volving in a circle, and to the force with which it tends to 
de from the centre of that circle; or if applied to the 
case of motion in any other curve, then it means the force 
with which the body tends to recede from the centre of the 
circle of curvature to its path at the point through which it 
ig, at any time, moving. When the body revolves in a cir- 
cular path, the circle of curvature to the path at anyone 
point evidently coincides with it throughout, and the chord 
of curvature becomes one of its diameters. Let the radius 
of the circle which the body thus describes be represented 
by R, then C = 2R; 



CENTRIFUGAL FORCE. 109 

■•■ ¥ =~w w 

Since in whatever curve a body is moving, it may be con 
ceived at any point of its path to be revolving in the circle 
of curvature to the curve at that point, the force F, with 
which it then tends to recede from the centre of the circle 
of curvature is represented by the above formula, R being 
taken to represent the radius of curvature at the point of its 
path through which it is moving. 

If a be the angular velocity of the body's revolution about 
the centre of its circle of curvature, then Y=aR, ; 

/.F=-a 2 R (102). 



122. From equation (100) we obtain 

Ftf 

Now (Art. 94.) — £ represents the additional velocity per 

second f which would be communicated to a body falling 
towards S, if the body fell freely and the force F remained 
constant. Moreover, by Art. 47. it appears, that V is the 
whole velocity which the body would on this supposition 
acquire, whilst it fell through a distance equal to JO, or to 
one quarter of the chord of curvature. Thus, then, the velo- 
city of a body revolving in any curve and attracted towards 
a centre of force is, at any point of that curve, equal to that 
which it would acquire in falling freely from that point to- 
wards the centre of force through one quarter of that chord 
of curvature which passes through the centre of force, if the 
force which acted upon it at that point in the curve re- 
mained constant during its descent. It is in this sense that 
the velocity of a body moving in any curve about a centre 
of force is said to be that due to one quarter the chord 

OF CURVATRE. 

123. The centrifugal force of a mass of finite dimensions. 



!S 



"** *•§•■ 




Let BC represenl a thin lamina or Blice 
of such a mass contained between two planes 
exceedingly near to one another, ana both 
perpendicular to a given axis A, about 
which the mass is maae to revolve. 



110 OENTKIFUGAL FORCE. 

Through A draw any two rectangular axes Ax and Ay, 
lei ///, be any element of the lamina whose weight is i#„ and 
Let AM, ana AN,, co-ordinates of ///,, he represented by #, 
and ;/,. Then by equation (102), if a represent the angular 
velocity of the revolution of the body, the centrifugal force 

mi the element m, is represented by — w,Ar/i,. Let now this 

. whose direction is Am, be resolved into two others, 
whose directions are Ax and Ay. The former will be repre- 

seiitedby — w.Am.. cos. xAm^ or by— w,x l9 and the latter 
9 9 

j „2 

} ) y — w l Am l cos. yAra,, or by— w 1 y 1 ; and the centrifugal 

" 9 9 

forces and all the other elements of the lamina being simi- 
larly resolved, we shall have obtained two sets of forces, 
those of the one 6et being parallel to Aa?, and represented 

J 2 3 

by _ w t x iy _ w a? , _ w s # 3 , &c. and those of the other set 
9 9 9 

2 2 2 

parallel to Ay represented by — i^y,, — w y 9 , — w 3 y 3 , &c. 

9 9 ' 9 

Now if X and Y represent the resolved parts parallel to 
the directions of Ax and Ay, of the resultant of these two 
sets of forces, then (Art. 11.) 



a* 



X=— w.x.+ — w„x.-i — wjxs % + . . . =— 2wx=— ¥G.; 
9 9 9 9 9 

*r a 2 a 2 a 2 a 2 a 2 ^ 

if G, and G 9 represent the co-ordinates AG, and AG, of the 
centre of gravity G of the lamina, and "W its weight 
(Art. 18A. 

Now the whole centrifugal force F on the lamina is the 
resultant of these two sets of forces, and is represented by 

\TJT (Art. 11.), 



/4 4 2 

-3WG; + ^W 2 G 3 3 = - W Vg; j + G 2 2 , or 
9 9 9 

F=jW.G ...... (103), 

where G- is taken to represent the distance AG of the centre 
of gravity of the lamina from the axis of revolution. 

Moreover, the direction of this resultant centrifugal force 



sc 



CENTRIFUGAL FORCE. Ill 

is through A, since the direction of all its components are 
through that point 



124:. From the above formula, it is apparent that if a body 
revolving round a fixed axis be conceived to 
be divided into laminae by planes perpendicu- 
lar to the axis, then the centrifugal force of 
each such laminae is the. same as it would 
have been if the whole of its weight had 
been collected in its centre of gravity ; so 
that if the centres of gravity of all the laminae 
be in the same plane passing through the 
axis, then, since the centrifugal force on each lamina has its 
direction from the axis through the centre of gravity of that 
lamina, it follows that all the centrifugal forces of these 
laminae are in the same plane, and that they are parallel 
forces, so that their resultant is equal to their sum, those 
being taken with a negative sign which correspond to 
laminae whose centres of gravity are on the opposite side of 
the axis from the rest, and whose centrifugal forces are 
therefore in the opposite directions to those of the rest. 
Thus if F' represent the whole centrifugal force of such a 




a 



mass, then F'= — 2WG. Now let W represent the weight 

of the whole mass, and G' the distance of its centre of gra- 
vity from the axis, therefore 2WG="W'G' ; 



j 

2 



.-. F'= -W'G' (104). 

9 
In the case, then, of a revolving body capable of being 
divided into laminae perpendicular to the axis of revolution, 
the centres of gravity of all of which laminae are in the 
same plane passing through the axis, the centrifugal force is 
the same as it would have been if the whole weight of the 
body had been collected in its centre of gravity, the same 
property obtaining in this case in respect to the whole body 
as obtains in respect to each of its individual lamina'. 
Since, moreover, the centrifugal forces upon the laminae are 
parallel forces when their centres of gravity arc all in the 
same plane passing through the axis of gravity, and Bince 
their directions are all in that plane, it Follows (Art. L6.), 
that if we take any point O in the axis, and measure the 
moments of these parallel forces from thai point, and call 
« the perpendicular distance OA of any lamina BC from 



112 THE l'i;::. VIRTUAL VELOCITIES. 

that point, and II the distance of their resultant from that 

point, then 

y y 
■• h= WgT « 

The eqnationa (104) and (105) determine the amount and 
the point of application of the resultant of the centrifugal 
forces upon the mass, upon the supposition that it can be 
divided into laminae perpendicular to the axis of revolution, 
all of which have their centres of gravity in the same plane 
passing through the axis. 

It is evident that this condition is satisfied, if the body be 
symmetrical as to a certain axis, and that axis be in the 
same plane with the axis of revolution, and therefore if it 
intersect or if it be parallel to the axis of revolution. 

If, in the case we have supposed, 2WG=0, that is, if the 
centre of gravity be m the axis of revolution, then the cen- 
trifugal force vanishes. This is evidently the case where a 
body revolves round its axis of symmetry. 



125. If the centres of gravity of the laminae into which 
the body is divided by planes perpendicular to 
the axis of revolution be not in the same plane 
(as in the figure), then the centrifugal forces of 
the different laminae will not lie in the same 
plane, but diverge from the axis in different 
directions round it. The amount and direction 
of their resultant cannot in this case be deter- 
mined by the equations which have been given 
above. 



The Principle of virtual Velocities. 

126. If any pressure P, whose point of application A is 
made to move through tht straight line AB, he resolved 
into three others X. V. Z, in the directions of the three 
rectangular axes, Ox, ()//, Qz ; and if AC, AD, and AE, 
be tht projections of A B upon th . then the work of 

P through AJB is equal to the sum of the works ofX, Y, Z, 
through AC, AD, and AE respectively, or X . AC-f-Y . 

AD + Z. AE=P. AM. 




THE PEINCIPLE OF YIKTUAX VELOCITIES. 113 



; 
o; 



Let the inclinations of the direction 

I i"W/f} of P to the axes Ox, Oy, Oz respec- 

! 'o tively, be represented by a, P, y, and 

the inclniations of AB to the same 

-a axes by « 1S /3 l5 y 13 

f "' 

/. (Art. 12.) X=P cos. «, Y=P cos. j8, Z=P cos. y; also AC 

=AB cos. a i5 AD=AB cos. /3 1} AE=AB cos. y 15 
/. X. AC=P. AB cos. a cos. « 15 Y. AD=P. AB cos. £ cos. ft, 

Z . AE=P . AB cos. y cos. y 13 
.-.X . AC+Y . AD + Z . AE=P . AB {cos. a cos. e^-f cos. £ 
cos. ^j + cos. y cos. y.J. 
But by a well-known theorem of trigonometry, cos. a cos. 
aj + cos. /3 cos. ft + cos. y cos. y 1 = cos. PAB, 

.-. X . AC+Y . AD + Z . AE=P . AB cos. PAB ; 
but AB cos. PAB = AM; 

.-.X . AC+Y . AD + Z . AE=P . AM. 

But (Art. 52.) the work of P through AM is equal to its 
work through AB. Therefore, &c* 

127. If any number of forces he in equilibrium {being in 
any way mechanically connected with one another), and if, 
subject to that connection, their different points of appli- 
cation be made to move, each through any exceedingly small 
distance, then the aggregate of the worlc of those forces, 
whose points of application are made to move towards the 

* This proposition may readily be deduced from Art. 53., for pressures equal 
and opposite to X, Y, Z, would just be in equilibrium with P, and these tend- 
ing to move the point A in one direction along the line AB, P tends to move 
it in the opposite direction, therefore in the motion of the point A through AB, 
the sum of the works of X, Y, Z, must equal the work of P. But the work of 
X, as its point of application moves through AB, is equal (Art. .V2.) to the* 
work of X through the projection of AB upon A*, thai is, through AC; simi- 
larly the work of Y, as its point of application moves through AB, is equal to 
its work through the projection of AH upon Ay, or through AD; and so of Z. 
The Bum of the works OI X, V, and Z, OS their point of application is made to 

move through AB, is therefore equal to what would have been the sum of their 
works had their points of application been made to move separately through 
AC, AD, AE; this last sum is therefore equal to the work of 1' through Al», 
which is equal to the work of P through AM, AM being the projection of ai; 
upon the direction of P. 

8 



1U 



I'KIX' H'l.K OP VIK1TAL VELOCITIES. 



,/,' al form \ I to them aoi t 

ahoU equal the aggregate of the work of thorn forces, the 
motions ofwhos< paints of application are opposed to the 
di f. the forces applii a to thi m. 

For let all the forces composing such a system be re- 
Bolved into three Beta of forces parallel to three rectangular 
axes, and let these three Bets <»f parallel forces be repre- 
sented by A, B, and G respectiveryi Tlien must the result- 
ant of the parallel forces of each set equal nothing. For if 
any of. these resultants had a finite value, then (by Art. 12.) 
the whole three sets of forces would have a resultant, which 
they cannot, since they are in equilibrium. 

Now let the motion of the points of application of the 
forces be conceived so small that the amounts and directions 
of the forces may be made to vary, during the motion, only 
by an exceedingly small quantity, and so that the resolved 
forces upon any point of application may remain sensibly 
unchanged. Also let ?/,. >/„. u„ represent the works of these 
resolved forces respectively on any point, and 2t*, the sum 
of all the works of the resolved forces of the set A, 2w 2 the 
sum of all the works of the forces of the set B, and St*, of the 
set C. Now since the parallel forces of the set A have no 
resultant, therefore (Art. 59.) the sum of the works of those 
forces of this set, whose points of application are moved 
towards the directions of their forces, is equal to the sum of 
the works of those whose points of application are moved 
from the directions of their forces, so that 2^=0, if the 
values of u x , which compose this sum, be taken with the 
positive or negative sign, according to the last mentioned 
condition. 

Similarly, 2t* t =0 and Zu 3 =Q, :. S^-H^ + ^^O. 

Now let U represent the actual work of that force P„ the 
works of whose components parallel to the three axes are 
represented by u„ w 2 , u i ; then by the last proposition, 

^+^ + ^3 = 11, 

.'. 2U=0 (106); 

in which don U is to be taken positively or negatively 

>rding to the same condition as it a , w a , u t ; that is, accord- 

is the w..rk at each point is done in the direction of the 

corresponding force, or in a direction opposite to it. Hence 

therefore it follows, from the above equations, that the sum 



THE PRINCIPLE OF VIS VTVA. 115 

of the works in one of these directions equals their sum in 
the opposite direction. Therefore, &c. 

The projection of the line described by the point of appli- 
cation of any force upon the direction of that force is called 
its virtual velocity, so that the product of the force by its 
virtual velocity is in fact the work of that force ; hence 
therefore, representing any force of the system by P, and 
its virtual velocity by p, we have Yp=JJ, and therefore, 
2Pp— 0, which is the principle of virtual velocities.* 

1.28. If there he a system of forces such that their points of 
application being moved through certain consecutive posi- 
tions, those forces are in all such positions in equilibrium, 
then in respect to any finite motion of the points of appli- 
cation through that series of positions, the aggregate of the 
work of those forces, which act in the directions in which 
their several points of application are made to move, is equal 
to the aggregate of the work in the opposite direction. 

This principle has been proved in the preceding proposi- 
tion, only when the motions communicated to the several 
points of application are exceedingly small, so that the work 
done by each force is done only through an exceedingly 
small space. It extends, however, to the case in which each 
point of application is made to move, and the work of each 
force to be done, through any distance, however great, pro- 
vided only that in all the different positions which the points 
of application of the forces of the system are thus made to 
take up, these forces be in equilibrium with one another ; for 
it is evident that if there be a series of such positions 
immediately adjacent to one another, then the principle 
obtains in respect to each small motion from one of this set 
of positions into the adjacent one, and therefore in respect 
to the sum of all such small motions as may take place in the 
system in its passage from any one position into any other, 
that is, in respect to the whole motion of the system through 
the intervening series of positions. Therefore, &c. 



The Principle of Yis Yiva. 

129. If the forces of any system be not in equilibrium with 
one another, then the difference between the aggregati work 

* This proof of the principle of virtual velocities is given here for the first 
time. 



116 THE PRINCIPLE OF VIS YIYA. 

of those whose tendency is in the direction of the motions 
of their several points of application, and those whose ten- 
dency is in the opposite direction, is equal to one half the 
aggregate vis viva of the system. 

In each of the consecutive positions which the "bodies com- 
posing the system are made successively to take up, let there 
he applied to each body a force equal to the effective force 
(Art. 103.) upon that body, but in an opposite direction; 
every position will then become one of equilibrium. 

Now, as the bodies which compose the system and the 
various points of application of the impressed forces move 
through any finite distances from one position into another, 
let 21^ represent the aggregate work of those impressed 
forces whose directions are towards the directions of the 
motions of their several points of application, and let 2w 2 
represent the work of those impyressed forces which act in the 
opposite directions ; also let 2u 2 represent the aggregate 
work of forces applied to the system equal and opposite to 
the effective forces upon it ; the directions of these forces 
opposite to the effective forces are manifestly opposite also 
to the directions of the motions of their several points of 
application, so that on the whole ^u x is the aggregate work 
of those forces whose directions are towards the motions of 
their several points of application, and lu 9 + 2w 3 the aggre- 
gate work opposed to them. Since therefore, by D'Alem- 
bert's principle, an equilibrium obtains in every consecutive 
position of the system, it follows by the last proposition, 
that 

.*. X^— 2to a =2w, (107). 

Now u s is taken to represent the work of a force equal and 
opposite to the effective force upon any body of the system ; 
but the work of such a force through any space is equal to 
the work which the effective force (being unopposed) accu- 
mulates in the body through that space (Art. 69.), or it is 
equal to one half the difference of the vires vivse of the body 
at the commencement and termination of the time during 
which that space is described (Art. 67.). Therefore 2w s 
equals one half the aggregate difference of the vires vwcb of 
the system at the two periods ; 

.-. Su t -2u % = -2w(v*-v;) (108). 



POSITION OF MAXIMUM OR MINIMUM VIS VIVA. HT 

Thus then it follows, that the difference between the aggre- 
gate work 2w x of those forces, the tendency of each of which 
is towards the direction of the motion of its point of applica- 
tion, and that 2u 2 of those the direction of each of which is 
opposed to the motion of its point of application (or, in other 
words the difference between the aggregate work of the 
accelerating forces of the system and that of the retarding 
forces), is equal to one half the vis viva accumulated or lost 
in the system whilst the work is being done, which is the 
Principle of Yis Yiva. 



130. One half the vis viva of the system measures its 
accumulated work; the principle of vis viva amounts, 
therefore, to no more than this, that the entire difference 
between the work done by those forces which tend to accele- 
rate the motions of the parts of the system to which they 
are applied, and those which tend to retard them, is accu- 
mulated in the moving parts of the system, no work 
whatever being lost, but all that accumulated which is dona 
upon it by the forces which tend to accelerate its motion, 
above that which is expended upon the retarding forces. 

This principle has been proved generally of any mechani- 
cal system; it is therefore true of the most complicate I 
machine. The entire amount of work done by the moving 
power, whatever it may be, upon that machine, is yielde 1 
partly at its working points in overcoming the resistances 
opposed there to its motion (that is, in doing its usefal 
work), it is partly expended in overcoming the friction an i 
other prejudicial resistances opposed to the motion of the 
machine between its moving and its working points, and all 
the rest is accumulated in the moving parts of the machine, 
ready to be yielded up under any deficiency of the moving 
power, or to carry on the machine for a time, should thj 
operation of that power be withdrawn. 



131. When the forces of any system {not in equilibrium in 
every position which the parts of that system may be 
made to take up) pass through a position of eqmlibrvum, 
the vis viva of the system, becomes a maximum or a 
minimum. 

For, as in Art. 129., let the aggregate work done in tin 4 
directions of the motions of the several parts of the BYStoxn 



118 POSITION OF MAXIMUM OR MINIMUM VIS YIYA. 

be represented by 2^ xJ and the aggregate work done in 
directions opposed to the motions of the several parts by 
2w 2 , then (Art. 129.), one half the acquired vis viva of 
system =2^ — 2^ 2 . Now as the system passes from any one 
position to any other, each of the quantities 2^ and 2?/ 2 is 
manifestly increased. If 2?^ increases by a greater quan- 
tity than 2^ 2 , then the vis viva is increased in this change 
of position ; if, on the contrary, it is increased by a less 
quantity than 2^ 2? then the vis viva is diminished. Thus if 
A2^ and A2^ 2 represent the increments of 2^ x and 2^ 2 in 
this change of position, then (2^ 1 + A2^ 1 )— (2^ 2 +A2^ 2 ) ? or 
(2u l — 2^ 2 ) + (a2'^ i — A2^ 2 ) 5 representing one half the vis 
viva after this change of position, it is manifest that the vis 
viva is increased or diminished by the change according as 
A2^ x is greater or less than A2^ 2 ; and that if A2u t be equa) 
to A2^ 2 then no change takes place in the amount of the 
vis viva of the system as it passes from the one position to 
the other. 

Now from the principle of virtual velocities (Art. 127.), 
it appears, that precisely this case occurs as the system 
passes through a position of equilibrium, the aggregate 
work of those forces whose tendency is to accelerate the 
motions of their points of application then precisely equal- 
ling that of the forces whose tendency is opposed to these 
motions. For an exceeding small change of position there- 
fore, passing through a position of equilibrium, A2^— A2^ 25 
an equality which does not, on the other hand, obtain, 
unless the body do thus pass through a position of equili- 
brium. 

Since then the sum 2^ — 2^ 2) and therefore the aggregate 
vis viva of the system, continually increases or diminishes 
up to a position of equilibrium, and then ceases (for a cer- 
tain finite space at least) to increase or diminish, it follows, 
that it is in that position a maximum or a minimum. 
Therefore, &c. 



132. When the forces of any system pass through a position 
of equilibrium, the vis viva becomes a maximum, or a 
minimum, according as that position is one of stable or 
unstable equilibrium. 

For it is clear that if the vis viva be a maximum in any 
position of the equilibrium of the system, so that after it 
has passed out of that position into another at some finite 






STABLE AND UNSTABLE EQUILrBEIUM. 119 

distance from it, the acquired vis viva may have become 
less than it was before, then the aggregate work of the 
forces which tend to accelerate the motion between these 
two positions must have been less than that of the forces 
which tend to retard the motion (Art. 131.). ISTow suppose 
the body to have been placed at rest in this position of 
equilibrium, and a small impulse to have been communi- 
cated to it, whence has resulted an aggregate amount of 
vis viva represented by 2mY 2 . In the transition from the 
first to the second position, let this vis viva have become 
%imf ; also let the aggregate work of the forces which have 
tended to accelerate the motion, between the two positions, 
be represented by sTJj, and that of the forces which have 
tended to retard the motion by 2U 2 ; then, for the reasons 
assigned above, it appears that 2U 2 is greater than sU^ 
Moreover, by the principle of vis viva, 

/. 2m^=2mY 2 -2(2U 2 -2U 1 ); 

in which equation the quantity 2(2U 2 — 2TJJ is essentially 
positive, in respect to the particular range of positions 
through which the disturbance is supposed to take place.* 

For every one of these positions there must then be a 
certain value of 2mV a , that is, a certain original impulse 
and disturbance of the system from its position of equili- 
brium, which will cause the second member of the above 
equation, and therefore its first member 2mv 2 , to vanish. 
Now every term of the sum Smv" is essentially positive ; 
this sum cannot therefore vanish unless each term of it 
vanish, that is, unless the velocity of each body of the 
system vanishes, or the whole be brought to rest. This 
repose of the system can, however, only be instantaneous ; 
for, by supposition, the position into which it has been dis- 
placed is not one of equilibrium. Moreover, the displac 
merit of the system cannot be continued in the direction in 
which it has hitherto taken place, for the negative term in 
the second member of the above equation would yel farther 
be increased so as to exceed the positive term, and the first 

* The disturbance is of course to be limited to that particular range of 
positions to which the supposed position of equilibrium Btands in the relation 
of a position of maximum vis viva, tf there be other positions o\' equiii* 
brium of the system, there will I"' other ranges of adjacent positions, in 

respect to each of which there obtains a similar relation of maximum Or mini- 
mum vis viva. 



BQUHJBBIUM. 

member s would thus become negative, which ia 
impossible. 

The in system can Then only I by 

the din :' the m - rtain of the elen 

which comix - _ , so that the i nding 

quantities by which 2U, and -I' , - ectively iner. 

may change their - _ -. and the whole quantity 2U,— ~\' 
which 1m tinnally may :.tinually 

Inish. This being i - a^ain 

until, when 2U, — 21^=0, it becomes a lal to SmV*; 

that is. until the system acq . - gain the via viva with 
which its disturbance commenced. 

Thus, then, it has been shown, that in respect to every 
one of the supposed \ - - aysi m* there is a cer- 

tain impulse <;»r amount of vis viva, which being communi- 
cated to the system when in equilibrium, will just cai 

- dilate as tar as that position, remain for an instant at 
rest in it. then return again towards its position of equili- 
brium, and re-acquire the vis viva with which its displace- 
ment commenced. Now this being true -ition 
of the Bupposed rang -'-ions, it follows that it is true 
disturbance or impulse which will not carry the 
system beyond this supposed range of p"siti - : - that, 
having 1 Laced by any such disturbance or impulse, 
the system will constantly return again towards the position 
of equilibrium from which it set out, and is stable in 
respect to that position. 

On the other hand, if the supposed position of equili- 
brium be one in which the vis viva is a minimum, then the 
__ gate work of the forces which tend to accelerate the 
•n must, after the system has passed through that - 
tion. that of the forces which tend to retard the 

motion; so that, adopting the same notation ;:- . ZlJ, 

must be greater than 2U,, and the second member of the 
equation essentially positive. "Whatever may have been the 
original impulse, and the communicated vis viva 2f»V*, 
- ' must UN continually increase: so that the whole 

mean never come instantam 

but on the contrary, the motions of its parts must continu- 
ously increase, and it must deviate continually farther from 
[uilibrium, in which position it can never 

* That is, of that range of position? over which the supposed position of 
equilibrium holds the relation of a position of maximum vis viva. 

+ Within that range of positions over which the supposed position of 
equilibrium holds the relation of minimum vis viva. 



DYNAMICAL STABILITY. 121 

rest. The position is thus one of unstable equilibrium. 
Therefore, &c. 

Dynamical Stability.* 

If a body be made, by the action of certain disturbing 
forces, to pass from one "position of equilibrium into another, 
and if in each of the intermediate positions these forces are 
in excess of the forces opposed to its motion, it is obvious 
that, by reason of this excess, the' motion will be continually 
accelerated, and that the body will reach its second position 
with a certain finite velocity, whose effect (measured under 
the form of vis viva) will be to carry it beyond that position. 
This however passed, the case will be reversed, the resist- 
ances will be in excess of the moving forces, and the body's 
velocity being continually diminished and eventually de- 
stroyed, it will, after resting for an instant, again return 
towards the position of equilibrium through which it had 
passed. It will not however finally rest in this position until 
it has completed other oscillations about it. Now the am- 
plitude of the first oscillation of the body beyond the posi- 
tion in which it is finally to rest, being its greatest ampli- 
tude of oscillation, involves practically an important condi- 
tion of its stability ; for it may be an amplitude sufficient to 
carry the body into its next adjacent position of equilibrium, 
which being, of necessity, a position of unstable equilibrium, 
the motion will be yet further continued and the body 
overturned. Different bodies requiring moreover different 
amounts of work to be done upon them to produce in all the 
same amplitude of oscillation, that is (relatively to that am- 
plitude) the most stable which requires the greatest amount 
of work to be so done upon it. It is this condition of stabi- 
lity, dependent upon dynamical considerations, to which, in 
the following paper, the name of dynamical stability is 
given. 

I cannot find that the question has before been considered 
in this point of view, but only in that which determines 
whether any given position be one of stable, unstable, <>r 
mixed equilibrium ; or which determines what pressure is 
necessary to retain the body at any given inclination from 
such a position. 

* Extracted from a paper "On Dynamical Stability, and on the Oscillations 
of Floating Bodies," by tin; author of this work, published in the Transactions 
of the Royal Society, Part. II. for 1850. The remainder of the paper will be 
found in the Appendix. 



PYNAMK AL STABILITY. 

1. To the discussion of the conditions of the dynamics^ 
Btabilil v of a body the principle of vis vwa readily lends 
itself. That principle,* when translated into a language 
which the labours of M. Ponoelet have made familiar to 
the uses of practical science, may be stated as follows: — 

u When, being acted upon by given forces, a body or sys- 
tezn of bodies has been moved from a state of rest, the differ- 
ence between the aggregate work of those forces whose 
tendencies are in the directions in which their points of 
application have been moved, and that of the forces whose 
tendencies are in the opposite direction, is equal to one-half 
the vis vwa of the system." 

Thus, if 2w, be taken to represent the aggregate work of 
the forces by which a body has been displaced from a posi- 
tioii in which it was at rest, and 2^ 2 the aggregate work 
(during this displacement) of the other forces applied to it ; 
and if the terms which compose 2^ and 2u., be understood 
to be taken positively or negatively, according as the ten- 
dencies of the corresponding forces are in the directions in 
which their points of application have been made to move 
or in the opposite directions ; then representing the aggre- 
gate vis viva of the body by - Zwv 2 . 

2^ + 2^= ^W, (1'). 

Now 2^ 2 representing the aggregate work of those forces 
which acted upon the body in the position from which it has 
been moved, may be supposed to the known; 2u Y may there- 
fore be determined in terms of the vis viva, or conversely. 

2. In the extreme position into which the body is made to 
oscillate and from which it begins to return, it, for an instant, 
rests. In this position, therefore, its vis viva disappears, and 
w r e have 

Zt^+2^0 (2'). 

This equation, in which 2w, and 2^ 2 are functions of the 
impressed forces and of the inclination, determines the ex- 
treme position into which the body is made to roll by the 
action of given disturbing forces; or, conversely, it deter- 
mines the force- by which it maybe made to roll into a 
given extreme position. 

* See Art. 129. 



DYNAMICAL STABILITY. 123 

3. The position in which it will finally rest is determined 
by the maximum value of 2^ + 2^ in equation (1') ; for, by 
a well-known property, the vis viva of a system* attains a 
maximum value when it passes through a position of stable, 
and a minimum, when it passes through a position of unstable 
equilibrium. The extreme position into which the body 
oscillates is therefore essentially different from that in which 
it will finally rest. 

4. Different bodies, requiring different amounts of work to 
be done upon them to bring them to the same given inclina- 
tion, that is (relatively to that inclination) the most stable 
which requires the greatest amount of work to be so done 
upon it, or in respect to which ^u x is the greatest. If, in- 
stead of all being brought to the same given inclination, each 
is brought into a position of unstable equilibrium, the corre- 
sponding value of ^u x represents the amount of work which 
must be done upon it to overthrow it, and may be considered 
to measure its absolute, as the former value measures its 
relative dynamical stability, f The absolute dynamical sta- 
bility of a body thus measured I propose to represent by the 
symbol U, and its relative dynamical stability, as to the 
inclination 6, by U(d). 

The measure of the absolute dynamical stability of a body 
is the maximum value of its relative stability, or U the max- 
imum of U(0) ; for whilst the body is made to incline from 
its position of stable equilibrium, it continually tends to 
return to it until it passes through a position of unstable 
equilibrium, when it tends to recede from it ; the aggregate 
amount of work necessary to produce this inclination must 
therefore continually increase until it passes through that 
position and afterwards diminish. 

5. The work opposed by the weight of a body to any 
change in its position is measured by the product of the 
vertical elevation of its centre of gravity by its weight.} 
Kepresenting therefore by "W the weight of the body, and 
by aH the vertical displacement of its centre of gravity 
when it is made to incline through an angle d, and observ- 
ing that the displacement^ this point is in a direction oppo- 
site to that in which the force applied to it acts, we nave 
2w 2 =— W.aH, and by equation (2'), 

* Art. 132. 

f It is obvious that the absolute dynamical stability of B body may be 
greater than that of another, whilst its stability, relatively to a given inclina- 
tion, is less; less work being required to incline it than the other at that 
angle, but more, entirely to overthrow it. 

\ Art. 60. 



124 iion. 

-W.aH=0 (3). 






If therefore do other force than its weight be opposed to a 
body's being overthrown, its absolute dynamical stability, 
when resting on a rigid surface, is measured by the product 

ightbythi height through which iU 
must d to bring it fi stable into an unstable 

posit juilibrium. 

vamical Stability of Floating Bodies. — The 
action of gnsts of wind upon a ship, or of blows of the sea, 
being measured in their effects upon it by their work, that 

1 is the most stable under the influence of these, or will 
roll and pitch the least (other things being the same), which 
requires the greatest amount of work to be done upon it to 
bring it to a given inclination; or, in respect to which the 
relative dynamical stability U (d) is the greatest for a given 
value of c 3 . In another sense, that ship may be said to be the 

-table which would require the greatest amount of work 
to be done upon it to bring it into a position from which it 
would not again right itself, or whose absolute dynamical 
stability U is the greatest. Subject to the one condition, 
the ship will roll the least, and subject to the other, it will 
be the Least likely to roll over. 

Thus the theory of dynamical stability involves a question 
of naval construction. It will be found discussed in its ap- 
plication to this question in the Appendix. 



FPJCTIOX. 

133. It is a matter of constant experience, that a certain 
resistance is opposed to the motion of one body on the sur- 
face of another under any pressure^ however smooth may be 
the surfaces of contact, not only at the first commencement, 
but at every subsequent period* of the motion; so that, not 
only is the exertion of a certain force necessary to cause the 
one body to pa— at first from a state of rest to a state of mo- 
tion upon the surface of the other, but that a certain force is 
further requisite * - - ate of motion. The resist- 

ance thus opposed to the motion of one body on the surface 
of another when the two are pressed together, is called trie- 



FRICTION. 125 

tion ; that which opposes itself to the transition from a state 
of continued rest to a state of motion is called the friction 
of quiescence ; that which continually accompanies the state 
of motion is called the friction of motion. 

The principal experiments on friction have been made by 
Coulomb*, Vince, G. Rennief, K WoodJ, and recently 
(at the expense of the French Government) by Morin.g 
They have reference, first, to the relation of the friction 
of quiescence to the friction of motion ; secondly, to the 
variation of the friction of the same surfaces of contact under 
different pressures ; thirdly, to the relation of the friction to 
the extent of the surface of contact ; fourthly, to the relation 
of the amount of the friction of motion to the velocity of the 
motion ; fifthly, to the influence of unguents on the laws of 
friction, and on its amount under the same circumstances of 
pressure and contact. The following are the principal facts 
which have resulted from these experiments ; they consti- 
tute the laws of friction. 

1st. That the friction of motion is subject to the same 
laws with the friction of quiescence (about to be stated), but 
agrees with them more accurately. That, under the same 
circumstances of pressure and contact, it is nevertheless dif- 
ferent in amount. 

2ndly. That, when no unguent is interposed, the friction 
of any two surfaces (whether of quiescence or of motion) is 
directly proportional to the force with which they are pressed 
perpendicularly together (up to a certain limit of that pres- 
sure per square inch), so that, for any two given surfaces 
of contact, there is a constant ratio of the friction to the per- 
pendicular pressure of the one surface upon the other 
Whilst this ratio is thus the same for the same surfaces of 
contact, it is different for different surfaces of contact. The 
particular value of it in respect to any two given surfaces 
of contact is called the CO-EFFICIENT of friction in re- 
spect to those surfaces. The co-efficients of friction in respect 
to those surfaces of contact, which for the most part form the 
moving surfaces in machinery, are collected in a table, which 
will be found at the termination of Art. 140. 

3rdly. That, when no unguent is interposed, the amount 
of the friction is, in every case, wholly independenl of the 
extent of the surfaces of contact, so that the force with which 
two surfaces are pressed together being the same, and 

* Mem. dcs Sav. Strang. 1781. t Phil - Tl ' illls - ls '-'-'- 

A Practical Treatise on Rail-roads, 8d ed. chap. 70. 
Mem. dc L'Institut. 1888, 1884, L888. 



12(5 FRICTION. 

M«»t exceeding a certain limit (per square inch), their friction 
is the same wn'atever may be the extent of their surfaces of 
contact. 

4thlv. That the friction of motion is wholly independent 
<>f the velocity of the motion.* 

5thly. That where unguents are interposed, the co-efficient 
of friction depends upon the nature of the unguent, and upon 
the greater or less abundance of the supply. In respect to 
the supply of the ungent, there are two extreme cases, that 
in which the surfaces of contact are but slightly rubbed with 
the unctuous matterf, and that in which, by reason of the 
ah in hint supply of the unguent, its viscous consistency, and 
the extent of the surfaces of contact in relation to the insist- 
ent pressure, a continuous stratum of unguent remains con- 
tinually interposed between the moving surfaces, and the 
friction is thereby diminished, as far as it is capable of being 
diminished, by the interposition of the particular unguent 
used. In this state the amount of friction is found (as might 
be expected) to be dependent rather upon the nature of the 
unguent than upon that of the surfaces of contact ; accord- 
ingly M. Morin, from the comparison of a great number of 
results, has arrived at the following remarkable conclusion, 
easily fixing itself in the memory, and of great practical 
value : — " that with unguents, hog s lard and olive oil, inter- 
posed in a continuous stratum between them, surfaces of wood 
on m^etal, wood on wood, metal on wood, and metal on metal 
(when in motion), have all of them very nearly the same co- 
efficient of friction, the value of that co-efficient being in all 
cases included between *07 and *08. 

" For the unguent tallow, the co-efficient is the same as for 
the other unguents in every case, except in that of metals upon 
metals. This unguent appears, from the experiments of Mo- 
rin, to be less suited to metallic substances than the others, 
and gives for the mean value of its co-efficient under the same 
circumstances -lO." 



134:. Whilst there is a remarkable uniformity in the results 
thus obtained in respect to the friction of surfaces, between 
which a perfect separation is effected throughout their whole 
extent by the interposition of a continuous stratum of the 

* This result, of so much importance in the theory of machines, is fully esta- 
blished by the experiments of Morin. 

f As, for instance, with an oiled or greasy cloth. 



FRICTION - . 127 

unguent, there is an infinite variety in respect to those states 
of unctuosity which occur between the extremes, of which 
we have spoken, of surfaces merely unctuous* and the most 
perfect state of lubrication attainable by the interposition 
of a given unguent. It is from this variety of states of the 
nnctnositv of rubbing surfaces, that so great a discrepancy 
has been found in the experiments upon friction with ungu- 
ents, a discrepancy which has not probably resulted so much 
from a difference in the quantity of the unguent supplied to 
the rubbing surfaces in different experiments, as in a diffe- 
rence of the relation of the insistent pressures to the extent 
of rubbing surface. It is evident, that for every description 
of unguent there must correspond a certain pressure per 
square inch, under wliich pressure a perfect separation of 
two surfaces is made by the interposition of a continuous 
stratum of that unguent between them, and which pressure 
per square inch being exceeded, that perfect separation can- 
not be attained, however abundant may be the supply of the 
unguent. 

The ingenious experiments of Mr. Nicholas "Wood-r, con- 
firmed by those of Mr. G. Rennie^:. have fully established 
these important conditions of the friction of unctuous surfaces. 
It is much to be regretted that we are in possession of no 
experiments directed specially to the determination of that 
particular pressure per square inch, which corresponds in 
respect to each unguent to the state of perfect separation, 
and to the determination of the co-efhcients of frictions in 
those different states of separation which correspend to pres- 
sures higher than this. 

It is evident, that where the extent of the surface sustain- 
ing a given pressure is so great as to make the pressure per 
square inch upon that surface less than that which corres- 
ponds to the state of perfect separation, this greater extent of 
surface tends to increase the friction by reason of that adhe- 
siveness of the unguent, dependent upon its greater or less 
viscosity, whose effect is proportional to the extent of the 
surfaces between which it is interposed. The experiments 
of Mr, Wood§ exhibit the effects of this adhesiveness in a 
remarkable point of view. 



* Or slightly rubbed with the unguent. 

f Treatise on Rail-roads, 3rd ed. p. 399. 

j Trans. Royal Soc. 1S29. 

g h is evident that, whilst by extending the unctuous surface which sustains 
any given pressure, we diminish the coefficient of friction up to a certain 
limit, we at the same time increase that adhesion of the surfaces which result* 



128 FRICTION. 

It is perhaps deserving of enquiry, whether in respect tc 
those considerable pressures under which the parts of the 
larger machines are accustomed to move upon one another, 
the adhesion of the unguent to the surfaces of contact, and 
the opposition presented to their motion by its viscidity^ are 
causes whose influence may be altogether neglected as com- 
pared with the ordinary friction. In the case of lighter 
machinery, as for instance that of clocks and watches, these 
considerations evidently rise into importance. 



135. The experiments of M. Morin show the friction of 
two surfaces which have been for a considerable time in con- 
tact, to be not only different in its anrbnnt from the friction 
of surfaces in continuous motion, but also, especially in this, 
thai the laws of friction (as stated above) are, in respect to 
the friction of quiescence, subject to causes of variation and 
uncertainty from which the friction of motion is exempt. 
This variation does not appear to depend upon the extent of 
the Burfaces of contact, in which case it might be referred to 
adhesion ; for with different pressures the co-efficient of the 
friction of quiescence was found, in certain cases, to vary 
exceedingly, although the surfaces of contact remained the 
same.* The uncertainty which would have been introduced 
into every question of construction by this consideration, is 
removed by a second very important fact developed in the 
course of the same experiments. It is this, that by the 
slightest jar or shock of two bodies in contact, their friction 
is made to pass from that state which accompanies quiescence 

from the viscosity of the unguent, so that there may be a point where the gain 
on the one hand begins to be exceeded by the loss on the other, and where 
the surface of minimum resistance under the given pressure is therefore 
attained. 

Mr. Wood considers the pressure per square inch, which corresponds to the 
minimum resistance, to be 90lbs. in the case of axles of wrought iron turning 
upon east iron, with fine neat's foot oil. The experiments of Mr. Wood, whilst 
they place the general results stated above in full evidence, can scarcely how- 
ever be considered satisfactory as to the particular numerical values of the con- 
stants Bought in this inquiry. In those experiments, and in others of the same 
class, the amount of friction is determined from the observed space or time 
through which a body projected with a given velocity moves before all its 
velocity is destroyed, that is, before its accumulated work is exhausted. This 
is an easy method of experiment, but liable to many inaccuracies. It is much 
to be regretted that the experiments of Morin were not extended to the fric- 
tion of unctuous surfaces, reference being had to the pressure per square 
inch. 

* Tims in the case of oak upon oak with parallel fibres, the co-efficient of 
the friction of quiescence varied, under different pressures upon the same sur- 
face, from •.">."> to 16. 



FRICTION. 129 

to that which accompanies motion ; and as every machine or 
structure, of whatever kind, may be considered to be subject 
to such shocks or imperceptible motions of its surfaces of 
contact, it is evident that the state of friction to be made 
the basis on which all questions of statics are to be deter- 
mined, should be that which accompanies continuous motion. 
The laws stated above have been shown, by the experiments 
of Morin, to obtain, in respect to that friction which accom- 
panies motion, with a precision and uniformity never before 
assigned to them ; they have given to all our calculations in 
respect to the theory of machines (whose moving surfaces 
have attained their proper bearings and been worn to their 
natural polish) a new and unlooked-for certainty, and may 
probably be ranked amongst the most accurate and valuable 
of the constants of practical science. 

It is, however, to be observed, that all these experiments 
were made under comparatively small insistent pressures as 
compared with the extent of the surface pressed (pressures 
not exceeding from one to two kilogrammes per square cen- 
timeter, or from about 14*3 to 28'6 lbs. per square inch.) In 
adopting the results of M. Morin, it is of importance to bear 
this fact in mind, because the experiments of Coulomb, and 
particularly the excellent experiments of Mr. Gr. Rennie, car- 
ried far beyond these limits of insistent pressure*, have fully 
shown the co-efficient of the friction of quiescence to increase 
rapidly, from some limit attained long before the surfaces 
abrade. In respect to some surfaces, as, for instance, wrought 
iron upon wrought iron, the co-efficient nearly tripled itself 
as the pressure advanced to the limits of abrasion. It is 

freatly to be regretted that no experiments have yet been 
irected to a determination of the precise limit about which 
this change in the value of the co-efficient begins to take 
place. It appears, indeed, in the experiments of Mr. Ren- 
nie in respect to some of the soft metals, as, for instance, tin 
upon tin, and tin upon cast iron ; but in respect to the harder 
metals, his experiments passing at once from a pressure of 
32 lbs, per square inch to a pressure of 1*66 cwt. per square 
inch, and the co-efficient (in the case of wrought iron for in- 
stance) from about '148 to '25, the limit which we seek is 
lost in the intervening chasm. The experiments of Mr. I Jen- 
nie have reference, however, only to the friction of qui- 
escence. It seems probable that the co-efficient of the tVie- 

* Mr. Rennie's experiments were carried, in some cases, to from 5 owt to 
? cwt. per square inch. 

9 



1 30 FRICTION. 

tion of motion remains constant under a wider range of pres- 
sure thai) that of quiescence. It is moreover certain, that 
the limits of pressure beyond which the surfaces of contact 
begin to destroy one another or to abrade, are sooner reached 
when «»nt' of them is in motion upon the other, than when 
they are at rest: it is also certain that these limits are not in- 
dependent of the velocity of the moving surface. The dis- 
cus8ion of this subject, as it connects itself especially with 
the friction of motion, is of great importance ; and it is to be 
regretted, that, with the means so munificently placed at his 
disposal by the French Government, M. Morin did not ex- 
tend his experiments to higher pressures, and direct them 
more particularly to the circumstances of pressure and velo- 
city under which a destruction of the rubbing surfaces first 
begins to show itself, and to the amount of the destruction 
of surface or wear of the material which corresponds to the 
same space traversed under different pressures and different 
velocities. Any accurate observer who should direct his 
attention to these subjects would greatly promote the inter- 
ests of practical science. 



Summary of the Laws of Friction. 

136. From what has here been stated it results, that if P 
represent the perpendicular or normal force by which one 
body is pressed upon the surface of another, F the friction of 
the two surfaces, or the force, which being applied parallel 
to their common surface of contact, w r ould cause one of them 
to slip upon the surface of the other, and/* the co-efficient of 
friction, then, in the case in which no unguent is interposed, 
f represents a constant quantity, and (Art. 133.) 

F=/P (109); 

a relation which obtains accurately in respect to the friction 
of motion, and approximately in respect to the friction of 
quiescence. 

137. The same relation obtains, moreover, in respect to 
unctuous surfaces when merely rubbed with the unguent, or 
where the presence of the unguent has no other influence 
than to increase the smoothness of the surfaces of contact 
without at all separating them from one another. 

In unctuous surfaces partially lubricated, or between which 



THE LIMITING ANGLE OF RESISTANCE. 131 

a stratum of unguent is partially interposed, the co-efficient 
of friction f is dependent for its amount upon the relation of 
the insistent pressure to the extent of the surface pressed, 
or upon the pressure per square inch of surface. This 
amount, corresponding to each pressure per square inch in 
respect to the different unguents used in machines, has not 
jet been made the subject of satisfactory experiments. 

The amount of the resistance F opposed to the sliding of 
the surfaces upon one another is, moreover, as well in this 
case as in that of surfaces perfectly lubricated, influenced by 
the adhesiveness of the unguent, and is therefore dependent 
upon the extent of the adhering surface ; so that, if S repre- 
sent the number of square units in this surface, and a the 
adherence of each square unit, then aS represents the whole 
adherence opposed to the sliding of the surfaces, and 

F=/P + aS (110); 

P 

where f is a function of the pressure per square unit ~-, and 

a is an exceedingly small factor dependent on the viscosity 
of the unguent. 



The limiting Angle of Resistance. 

We shall, for the present, suppose the parts of a solid body 
to cohere so firmly, as to be incapable of separation by the 
action of any force which may be impressed upon them. 
The limits within which this suposition is true will be dis- 
cussed hereafter. 

It is not to this resistance that our present inquiry has 
reference, but to that which results from the friction of the 
surface of bodies on one another, and especially to the direc- 
tion of that resistance. 



138. Any pressure applied to the surface of an immoveabl e 
solid body by the intervention of (mother bod;/ moveable 
upon it, will be sustained by the resistance of wa surface* 
of contact, whatever be its direction, provided only th< sin- 
gle which that direction makes with tin /» /■/>> ndioular to 
the surfaces of contact do not exceed a certain angh, called 
the limiting angle of resistance of thost Bl RI 




L32 THE LIMITING ANGLE OF RESISTANCE. 

This is true, however great the pressure may he. Also if 
the inclination of the pressure to the pi rj» ndioula/r exceed 
the In a Win ij angle of resistance, then this pressure will not 
he sustained hy the resistance of the surfaces of contact y 
and this is true, however small the pressure may he. 

represent the direction in which the surfaces of 
two bodies are pressed together at Q, and let 
QA be a perpendicular or normal to the sur- 
faces of contact at that point, then will the pres- 
sure PQ be sustained by the resistance of the 
surfaces, however great it may be, provided its 
direction lie within a certain given angle AQB, 
called the limiting angle of resistance ; and it will not be sus- 
tained, however small it may be, provided its direction lie 
without that angle. For let this pressure be represented by 
PQ, and let it be resolved into two others AQ and PQ, of 
which AQ is that by which it presses the surfaces together 
perpendicularly, and EQ that by which it tends to cause 
them to slide upon one another, if therefore the friction F 
produced by the first of these pressures exceed the second 
pressure PQ, then the one body will not be made to slip 
upon the other by this pressure PQ, however great it may 
be ; but if the friction F, produced by the perpendicular 
pressure AQ, be less than the pressure PQ, then the one 
body will be made to slip upon the other, however small PQ 
may be. Let the pressure in the direction PQ be repre- 
sented by P, and the angle AQP by d, the perpendicular 
pressure in AQ is then represented by P cos. d, and therefore 
the friction of the surfaces of contact by/T cos. 6, f repre- 
senting the co-efficient of friction (Art, 136.). Moreover, the 
resolved pressure in the direction PQ is represented by P 
sin. 6. The pressure P will therefore be sustained by the 
friction of the surfaces of contact or not, according as 

P sin. 6 is less or greater than f~P cos. & ; 

or, dividing both sides of this inequality by P cos. 6, ac 
cording as 

tan. 6 is less or greater than f. 

Let, now, the angle AQB equal that angle whose tangent is 
f and let it be represented by <f>, so that tan. <P=f. Substi- 
tuting this value of f in the last inequality, it appears that 
the pressure P will be sustained by the friction of the sur- 
faces of contact or not, according as 



THE TWO STATES BORDERING UPON MOTION. 133 

tan. 6 is greater or less than tan. 0, 
that is, according as 

6 is less or greater than 0, 
or according as 

AQP is less or greater than AQB. 
Therefore, &c. [q. e. d.] 

The Cone of Resistance. 

139. If the angle AQB be conceived to revolve about the 
axis AQ, so that BQ may generate the surface of 
A\ a. a cone BQC, then this cone is called the cone op 
s / )\\v resistance : it is evident, that any pressure, how- 
Q \4*£r^K ever great, applied to the surfaces of contact at 
\ Q will be sustained by the resistance of the sur- 

faces of contact, provided its direction be any 
where within the surface of this cone ; and that it will not 
be sustained, however small it may be, if its direction lie any 
where without it. 



The Two States bordering upon Motion. 

140. If the direction of the pressure coincide with the sur- 
face of the cone, it will be sustained by the friction of the 
surfaces of contact, but the body to which it is applied will 
be upon the point of slipping upon the other. The state of 
the equilibrium of this body is then said to be that border- 
ing upon motion. If the pressure P admit of being applied 
in any direction about the point Q, there are evidently an 
infinity of such states of the equilibrium bordering upon mo- 
tion, corresponding to all the possible positions of P on the 
surface of the cone. 

If the pressure P admit of being applied only in the same 
plane, there are but two such states, corresponding to those 
directions of P, which coincide with the two intersections of 
this plane with the surface of the cone ; these are called the 
superior and inferior states bordering upon motion. In the 
case in which the direction of Pis limited to the plane AQI», 
BQ and CQ represent its directions corresponding to the 



134 THE TWO STATES BORDERING UPON MOTION. 

two states bordering on motion. Any direction of P within 
the angle BQC corresponds to a state of equilibrium; any 
direction, without this angle, to a state of motion. 



141. Since, when the direction of the pressure P coincides 
with the surface of the cone of resistance, the equilibrium is 
in the state bordering upon motion ; it follows, conversely, 
and for the same reasons, that this is the direction of the 
pressure sustained by the surfaces of contact of two bodies 
whenever the state of their equilibrium is that bordering upon 
motion. This being, moreover, the direction of the pressure 
of the one body upon the other is manifestly the direction of 
the resistance opposed by the second body to the pressure of 
the first at their surface of contact, for this single pressure 
and this single resistance are forces in equilibrium, and there- 
fore equal and opposite. All that has been said above of the 
single pressure and the single resistance sustained by two 
surfaces of contact, is manifestly true of the resultant of any 
number of such pressures, and of the resultant of any num- 
ber of such resistances. Thus then it follows, that when any 
wwriber of pressures applied to a body moveable upon another 
ichich is fixed, are sustained by the resistance of the surfaces 
of contact of the two bodies, and are in the state of equilibrium 
oordering upon motion, then the direction of the resultant of 
these pressures coincides with the surface of the cone of resist- 
ance, as does that also of the resultant of the resistances of the 
(I [f * rent points of the surfaces of contact*, that is, they are 
both inclined to the perpendicular to the surfaces of contact 
(at the point where they intersect it), at an angle equal to the 
limiting angle of resistance. 

* The properties of the limiting angle of resistance and the coiu> o r inst- 
ance, were first given by the author of this work in a paper published \n the 
Cambridge Philosophical Transactions, vol. v. 



FRICTION. 



135 



Table I. 
Friction of Plane Surfaces, when they have been some time in Co t'.act. 



Surfaces in Contact. 



Disposition of 
the Fibres. 



Experiments of M. Morix. 



Oak uDon oak 



Oak upon elm 
Elm upon oak 



Ash, fir, beech, service- ) 
tree, upon oak 



Tanned leather upon oak^ 



Black 
dressed 
leather, 
or strap 
leather 



upon a plane J 
surface of ' 
oak 

upon a round- 
ed surface 
of oak 






Hemp matting upon oak 
Hemp cords upon oak - 

Iron upon oak 

Cast-iron upon oak 
Copper upon oak - 



Ox-hide as a piston sheath ) 
upon cast-iron ) 



parallel -j 

ditto -j 

perpendicu- j 
lar ( 

ditto 

endways of " 
one upon 
the flat 
surface of 
the other ^ 

parallel 

ditto 

ditto 

perpendicu- 
lar 

parallel 

the leather 
flat 

the leather" 
length- 
ways, but 
sideways 

parallel 

perpendicu- 
lar 

parallel 

ditto 

ditto 



ditto 

ditto 
ditto 

flat or side- 
ways 



State of the 
Surfaces. 



without 
unguent 

rubbed with 
dry soap 

without 
unguent 

with water 

without 
unguent 

ditto 
ditto 
rubbed with 

dry soap 
without 

unguent 

ditto 
ditto 

ditto 

steeped in 
water 

without 
unguent 

ditto 

ditto 
steeped in 

water 
without 

unguent 
ditto 
steeped in 

water 
ditto 
without 

uuguent 
steeped in 

water 
with oil, 

tallow, or 

hog's la i >l 



Coefficient 
of Friction. 



0-62 

0-44 

0-54 
0-71 

0*43 



0-38 
0-69 



0-74 

0'47 

0-50 
0-87 

0-80 
0-62 
0-65 
0-65 
0-62 

0*68 
0*12 



Limiting 
Angle of 
Resist- 
ance. 



31° 48' 

23 45 

28 22 
35 23 

23 16 



20 49 

34 37 



0-41 


22 


18 


0-57 


29 


41 


0-53 


27 


56 


0'61 


31 


23 


0-43 


23 


16 


0-79 


38 


19 



36 30 

25 11 

26 34 
41 2 

38 40 

31 48 

33 2 

33 2 

31 48 

81 48 

6 51 



136 



Ki:i«:TION. 



Surfaces In (J 



, or ) 



BZPXBIMKNTS Of M. MORIN. 

— continued. 

Black dressed leather, 

strap leather, upon 
cast-iron pulley 

Oast-iron upon cast-iron - 

Iron upon cast-iron 
Oak, elm, yoke elm, iron, 

east-iron, and braes 

sliding two and two, 

one upon another 
Calcareous oolite stone 

upon calcareous oolite 
Hard calcareous stone, 

called muschelkalk, 

upon calcareous oolite 
Brick upon calcareous ' 

oolite 
Oak upon calcareous 

oolite i 

Iron upon calcareous oolite 

Hard calcareous stone, or ) 

muschelkalk, upon > 

muschelkalk ) 

Calcareous oolite stone ) 

upon muschelkalk j 

Brick upon muschelkalk - 

Iron upon muschelkalk - 

Oak upon muschelkalk - 



Calcareous oolite stone 
upon calcareous oolite 



Disposition of 
the Fibres. 



flat 

ditto 
ditto 

ditto 

ditto 
ditto 

ditto 

wood end- 
ways 

flat 
ditto 

ditto 

ditto 
ditto 
ditto 



ditto 



State of the 
Surfaces. 



Co efficient 
of Friction 



without 

unguent 
steeped 
without 

unguent 
ditto 

with tallow 

with oil, or 

hog's lard 

without 
unguent 

ditto 



ditto 
ditto 
ditto 

ditto 

ditto 

ditto 
ditto 
ditto 

with a coat- 
ing of mor- 
tar, of three 
parts of fine 
sand and 
one part of 
slack lime 



0-28 
0-38 
0-16 
019 

o-iof 

0-15$ 

o-74 

0-75 

0-67 
0-63 
0-49 

0-70 

0-75 

0-67 
0'42 
0-64 



0'74§ 



Ltiaitlng 

Angle of 
Reals t- 



15° 39 

20 49 

9 6 

10 46 

5 43 

8 32 

36 30 

36 52 

33 50 

32 13 

26 7 

35 



36 52 

33 50 

22 47 

32 38 



86 80 



* The surfaces retaining some unctuousness. 

f When the contact has not lasted long enough to express the grease. 
\ When the contact has lasted long enough to express the grease and brir.g 
back the surfaces to an unctuous state. 

§ After a contact of from ten to fifteen minutes. 



FRICTION. 



137 



Nature of Bodies and Unguents. 


Co-efficient 
ofFriction. 


Limiting 
Angle. 


Soft calcareous stone, well dressed, upon the same 

Hard calcareous stone, ditto - 

Common brick, ditto - 

Oak, endways, ditto - - - - 

Wrought iron, ditto - 

Hard calcareous stone, well dressed, upon hard calcare- 
ous stone -.--.. 

Soft, ditto ...... 

Common brick, ditto ..... 

Oak, endways, ditto ..... 

Wrought iron, ditto ..... 

Soft calcareous stone upon soft calcareous stone, with 
fresh mortar of fine sand - - - - 

Experiments by Different Observers. 

Smooth free-stone upon smooth free-stone, dry (Rennie) 
Ditto, with fresh mortar (Rennie) ... 
Hard polished calcareous stone upon hard polished cal- 
careous stone ------ 

Calcareous stone upon ditto, both surfaces being made 
rough with a chisel (Bonchardi) ... 
Well dressed granite upon rough granite (Rennie) 
Ditto, with fresh mortar, ditto (Rennie) - - - 
Box of wood upon pavement (Regnier) - - - 
Ditto upon beaten earth (Herbert) ... 
Libage stone upon a bed of dry clay 
Ditto, the clay being damp and soft . - - 
Ditto, the clay being equally damp, but covered with 
thick sand (Greve) - - - - - 

- 


0-74 
0-75 
0-67 
0-63 
0-49 

0-70 
0-75 
0-67 
0-64 
0*42 

0-74 

0-71 
0-66 

0-58 

0-78 
0-66 
0-49 
0-58 
0-33 
0'51 
0-34 

0*40 


36° 30' 
36 52 
33 50 

32 13 
26 7 

35 

36 52 

33 50 

32 37 

22 47 

36 30 

35 23 

33 26 

30 7 

37 58 
33 26 

26 7 
30 7 
18 16 

27 2 
18 47 

21 48 



138 



FiilCTION. 



Table II. 
Friction of Plane Surfaces, in Motion one upon the other. 



Surfaces in Contact. 



Experiments of M. Morin, 



Oak upon oak 



Elm upon oak 

Ash, fir, beech, wild pear- 
tree, and service-tree, 
upon oak 



Iron upon oak 



Cast-iron upon oak 

Copper upon oak - 

Iron upon elm 
Cast-iron upon elm - 
Black dressed leather ) 
upon oak ) 

Tanned leather upon oak \ 



Tanned leather upon 
cast-iron and brass 



Disposition of 
the Fibres. 



parallel 

ditto 

perpendicu- j 
lar 



ditto 
wood 



end- 



ways on 
wood 
length- 
ways 

parallel 

perpendicu- 
lar 

parallel 

ditto 



ditto 



ditto 



ditto 

ditto 
ditto 

ditto 



flat, or 
length- 
ways, and 
edgeways 



ditto 



State of the 
Surfaces. 



without 

unguent 
rubbed with 

dry soap 
without 

unguent 
steeped in 

water 

without 
unguent 



ditto 
ditto 
ditto 

ditto 

ditto 

with water 
rubbed with 

dry soap 
without 

unguent 
with water 
rubbed with 

dry soap 
without 

unguent 
ditto 
ditto 

ditto 



ditto 
with water 

without 

unguent 
steeped in 

water 
greased and 

steeped in 

water 
with oil 



Co efficient 
of Friction 



0-48 
0-16 
0-34 
0-25 

0-19 

0-43 

0-45 

0-25 

0-36 to 
0-40 

0-62 
0-26 

0-21 

0-49 
0-22 
0-19 

0-62 

0-25 
0-20 

0-27 

3-30 to 

0-35 

0-29 

0-56 

0-36 

0-23 
0-15 



Limiting 
Angle of 
BealBt- 



25 c 39' 
9 6 

18 4* 
14 3 

10 46 

23 17 

24 14 
14 3 

19 48 
21 49 

31 48 
14 35 

11 52 

26 7 

12 25 

10 46 

31 48 

14 3 

11 19 

15 7 

16 42 
19 18 
16 11 

29 15 

19 48 

12 58 
8 32 



FRICTION. 



139 



r 

Surfaces in Contact. 


Disposition of 


State of the 


Coefficient 


Limiting 
Angle of 




the Fibres. 


Surfaces. 


of Friction. 


Resist- 
ance. 


Experiments op M. Morin. 








i 


— continued. 










Hemp, in threads or in J 


parallel -j 


without ) 
unguent \ 


0-52 


27°29' 


cord, upon oak 


perpendicu- ) 
lar j" 

parallel ■< 


with water 


0-33 


18 16 


/ Oak and elm upon cast- \ 
iron j 


without ) 
unguent j" 


0-38 


20 49 


Wild pear-tree, ditto 


ditto 


ditto 


0-44 


23 45 


Iron upon iron 


ditto 


ditto 


0-44 


23 45* 


Iron upon cast-iron and ) 
brass J 


ditto 


ditto 


0-18f 


10 13 


Cast-iron, ditto 


ditto 


ditto 


0-15 


8 32 


C upon brass - 


ditto 


ditto 


0-20 


11 19 


Brass < upon cast-iron 


ditto 


ditto 


0-22 


12 25 


( upon iron - 


ditto 


ditto 

greased in" 
the usual 


0-16^: 


9 6 


Oak, elm, yoke elm, wild ' 
pear, cast-iron, wrought 
iron, steel, and moving 




ditto 


way with 
tallow, y 
hog's lard, 
oil, soft 


0-07 to 
0-08§ 


U 1 

(4 35 


one upon another, or on 










themselves 






gom 
slightly 










greasy to >• 


0-15 


8 32 






the touch ) 






Calcareous oolite stone ) 
upon calcareous oolite ) 


ditto -j 


without ) 
unguent \ 


0-64 


82 37 


Calcareous stone, called ) 










muschelkalk, upon cal- >• 


ditto 


ditto 


0-67 


33 50 


careous oolite ) 










Common brick upon cal- ) 
careous oolite \ 


ditto 


ditto 


0-65 


33 2 


Oak upon calcareous j 
oolite ] 


wood end- ) 
ways ) 


ditto 


0-38 


20 49 


Wrought iron, ditto 


parallel 


ditto 


0-69 


34 37 


Calcareous stone, called } 










muschelkalk,upon mus- >• 


ditto 


ditto 


0-38 


20 49 


chelkalk ) 










Calcareous oolite stone ) 
upon muschelkalk [ 


ditto 


ditto 


0-65 


33 2 


Common brick, ditto 


ditto 


ditto 


0-60 


30 58 i 


Oak upon muschelkalk ■< 


wood end- ) 

ways \ 


ditto 


0-38 


20 49 




ditto 


0-24 




Iron upon muschelkalk - 


parallel < j saturated ) 
( with water ) 


0-30 


16 42 



* The surfaces wear when there is no grease. 

+ The surfaces still retaining a little unctuousness. \ Ibid. 

§ When the grease is constantly renewed and uniformly distributed, this 
proportion can be reduced to 0*05. 



14U 



noH. 



Table III. 
■ of Gudgeon* or AxU-tnds, in Motion, upon their Bearings. 
m the experiments of Morin.) 



hi Crv -.:-.. 



- .■-. : ■ ■ ;-.-:. ;• 



Co-efficient of Friction 
the G: 



In Ike 



Continuously. 



Cast-iron axles 
in cast-iron 
bearings 



ditto 



ajle; 



Cast-iron axles 
in lignum vita 
bearings 



Wrought-iron 
axle- 
iron bearings 



Iron axles in 
brass bearings 



coated with oil of 
olives, with hog's 
lard, tallow, and 
:om 

with the same, 
and water 

coated with as- 
phaitum 

greasy 

greasy and wetted 

coated with oil of 
olives, with hog's 
lard, tallow, and 
;om 

and damped 
scarcely greasy 
without unguent 
with oil or hog's 

lard 
greasy with ditto 
greasy, with a | 

mixture of hog's ; 

lard and molyb- 

daena 
coated with oil 

of olives, tallow, 

hog's lard, or 
:om 
coated with oil of 

olives, hog's lard, 

or tallow 
coated with hard 






3 



o-u 

0-14 



::■: 

0-16 
I 

0-19 
0-18 






i 



\ 

isy and wetted 
- 
Iron axles in ) coated with oil ) 
lignum viue J- or hog's lard 
bearings * . asy 

Brass axles in i coated with oil 
brass bearings \ with hog's lard 
Brass axles . , ., . 

-:ronbear-f 00ate * ^ od '. 
ings * or toUo,r 



0*09 
019 

0-11 

: 






Sfl 



0-19 






0-090 









L - ..• - z 
of 



4 35 
3 6 

3 6 



- 


oS 


- 


•:> 


4 





4 


■ -■ 


1 


1 


1 


1 


9 


8 


10 M 


10 12 


5 


1 


5 43 



: N 



4 




4 35 


3 


1 


4 




4 35 


3 


B 


5 


9 


10 46 


14 


. 



to 0-052 



10 46 

5 9 
2 35 



* The surfaces beginning to wear. 



FRICTION. 



141 



Surfaces in Contact. 


State of the Surfaces. 


Co-efficient of Friction when 
the Grease is renewed. 


Limiting 


In the usual 
Way. 


Continuously. 


Resistance. 


Lignum vitae \ 
axles, ditto j 

Lignum vitae "1 
axles in lig- 1 
num vitae f 
bearings J 


coated with hog's 

lard 
greasy- 
coated with hog's 

lard 


1 0-12 
0-15 

I- " 


0-07 


6°51' 
8 82 

4 



Table IY. 

Co-efficients of Friction under Pressures increased continually up to the 

Limits of Abrasion. 

(From the experiments of Mr. G. Rennie.*) 



Pressure per 


Co-efficients of Friction. 










Square Inch. 


Wrought-iron 


Wrought-iron 


Steel upon 


Brass upon 














Wrought-iron. 


Cast-iron. 






32- 5 lbs. 


•140 


•174 


•166 


•157 


1-66 cwt. 


•250 


•275 


•300 


•225 


2-00 


•271 


•292 


•333 


•219 


2-33 


•285 


•321 


•340 


•214 


2-66 


•297 


•329 


•344 


•211 


3-00 


•312 


•333 


•347 


•215 


3-33 


•350 


•351 


•351 


•206 


3-66 


•376 


•353 


•353 


•205 


4-00 


•376 


•365 


•354 


•208 


4-33 


•395 


•366 


•356 


•221 


4-66 


•403 


•366 


•357 


•223 


5-00 


•409 


•367 


•358 


•233 


5-33 




•367 


•359 


•234 


5-66 




•367 


•367 


•235 


6-00 




•376 


•403 


•233 


6-33 




•434 




•234 


6-6C 








•235 


7-00 








•232 


733 








•273 



* Phil. Trans. 1829, table 8. p. 159. 



14-2 THE KIGIDITY OF CORD8. 




THE RIGIDITY OF CORDS. 

142. It is evident that, by reason of that resistance t: 
deflexion which constitutes the ri- 
gidity of a cord, a certain force or 
pressure must be called into action 
whenever it is made to change its 
rectilineal direction, so as to adapt 
itself to the form of any curved sur- 
face over which it is made to pass ; 
as, for instance, over the circumfe- 
rence of a pulley or wheel. Sup- 
pose such a cord to sustain tensions represented by P, and 
JP 2 , of which P, is on the point of preponderating, and let 
the friction of the axis of the pulley be, for the present, 
neglected. It is manifest that, in order to supply the force 
necessary to overcome the rigidity of the cord and to pro- 
duce its deflection at B, the tension P, must exceed P 2 ; 
whereas, if there were no rigidity, P, would equal P 2 ; so 
that the effect of the rigidity in increasing the tension P, is 
the same as though it had, by a certain quantity, increased 
the tension P 2 . Now, from a very numerous 6eries of 
experiments made by Coulomb upon this subject, it appears 
that the quantity by which the tension P 2 may thus be con- 
sidered to be increased by the rigidity, is partly constant 
and partly dependent on the amount of P 2 ; so as to be 
represented by an algebraical formula of two terms, one 
of which is a constant quantity, and the other the product 
of a constant quantity by P 2 . Thus if D represent the 
constant part of this formula, and E the constant factor 
of P 2 , then is the effect of the rigidity of the cord the same 
as though the tension P 2 were increased by the quantitv 
D + E.P, 

AVlien the cord, instead of being bent, under different 
pressures, upon circular arcs of equal radii, was bent upon 
circular arcs of different radii, then this quantity D + E . P a ; 
by which the tension P 2 may be considered to be increased 
by the rigidity, was found to vary inversely as the radii 
of the arcs; so that, on the whole, it may be represented 
by the formula 



THE KIGIDITY OF COEDS. 143 

D+E . P n 



E 



(HI), 



where E represents the radius of the circular arc over which 
the rope is bent. Thus it appears that the yielding tension 
P 2 may be considered to have been increased by the rigidity 
of the rope, when in the state bordering upon motion, so as 
to become 

r a + R 

This formula applies only to the bending of the same cord 
under different tensions upon different circular arcs : for dif- 
ferent cords, the constants D and E vary (within certain 
limits to be specified) as the squares of the diamsters or of the 
circumferences of the cords, in respect to new cords, wet or 
dry ; in respect to old cords they vary nearly as the power f 
of the diameters or circumferences. 

Tables have been furnished by Coulomb of the values of 
the constants D and E. These tables, reduced to English 
measures, are given on the next page.* 

* The rigidity of the cord exerts its influence to increase resistance only at 
that point where the cord winds upon the pulley ; at the point where it leaves 
the pulley its elasticity favours rather, and does not perceptibly affect, the 
conditions of the equilibrium. 

In all calculations of machines, in which the moving power is applied by the 
intervention of a rope passing over a pulley, one-half the diameter ef rope is 
to be added to the radius of the pulley, or to the perpendicular on the direction 
of the rope from the point whence the moments are measured, the pressure 
applied to the rope producing the same effect as though it were all exerted 
along the axis of the rope. 



144 



THE lilGIDITT OF CORDS. 



Table V. Rigiditt of Ropes. 



Table of the ralues of the constant* D and E, according to the experiments of 
Coulomb {reduced to English measures}. The radius R of the pulley is to U 
taken in 



New dry cords. Rigidity proportional to the square of the 



Circumference of 
the Rope in Inches. 



Value .: Z ml 



. 



B-413702 



" 17 J 




Hew ropes dipped in water. Rigidity proportional \2T 
he square of the circumference. 



Circumference of 
the Rope in Inches. 



Value of D in lbs. \ Talue of E in lbs. 



1 




i -__■.- 

16*835606 


- 
• - 1755 



ar 


■*M 


. 


1 •- 


ii 


1-91 


i-j 


:•« 


it 


lf« 


M 


1-9C 


l-C 


m 


!"• 


!» 


• 


i r, 


1-8 


S-M 


l-» 


ra 


: 


* 



Dry half-worn ror :ity proportional to the square root 

of the cube of the circumferer. [ 



i - r •- ■' : - 

of L 

tioos of the Li- 



Circumference of 
the Rope in Inches. 


Talue of D in lbs. Yalae of E in lbs. 


1 
- 
4 

8 


-"- 
•41c ; - -_- 
-115 




No. 4. Wetted half-worn cords. Rigidity proportional 
to the square root of the cube of the circumference. 


St 


7 - -. : 


Circumference of ' v , nfl , ,. . .. 
the Rope in Inches j Talue of D m Ib " 


Value of E in lbs. 1 


. 

ii 

i-» 

1-S 
W 

1-t 


11M 

:-,-. 

1-3T , 

: M 

- . ' 

; Bl 


1 

4 - 

8 6-61 - 


■006401 

■ois: " 

•l44-_. 

. i 







THE KIGIDITY OF COEDS. 14! 

Xo. 5. Tarred rope. Rigidity proportional to the number of strands. 



Number of Strands. 


Value of D in lbs. 


Value of E in lbs. 


6 

15 
30 


0-33390 
0-17212 
1-25294 


0-009305 
0-021713 
0-044983 



To determine the constants D and E for ropes whose circumferences are 
intermediate to those of the tables, find the ratio of the given circumference 
to that nearest to it in the tables, and seek this ratio or proportion in the first 
column of the auxiliary table to the right of the page. The corresponding 
number in the second column of this auxiliary table is a factor by which the 
values of D and E for the nearest circumference in the principal tables being 
multiplied, their values for the given circumference will be determined.* 

* Note (s) Ed. App. 



10 






146 THE THEOBT OF MACHINES. 



P A. PI T III. 
THE THEORY OF MACHI 



14. .its of a machine are divisible into those which 

- - - -hose 

which Oj- wnediatdy wpan the work to be performed, 

and * - .Jch com m tween the two, or which 

conduct the pow^r or in the moving to the working 

points of the machine. The first class may be called ei 
ebs. the second opee. 1 the third <x>ionrsiCAT«: :-. 

work. 

- :: — : :■" : " T " bk by ^f aches 

144. The moving p:»wer divides itself whilst it operates in 
a machiEv. _ . Into that which overcomes tL 

:' the machine, or those which are opposed by 
friction and other causes uselessly absorbing the work in its 
transmission. Secondly. Into that which a 

n of the various moving parts of the machine : so long 

as the work done by the moving power up a that 

expended upon the varioi:- -othem 

of the machine <Art. 129. . 2! Into that which 

comes th. f a, .vhich are opposed to 

the motion of the machine at the working point or points 

by the useful work which is to be done by it. Thus, then. 

ork done by the moving power upon the moving points 

of the m; as distinguished from the working points > 

divides itself in the act of transmission, first. Into the work 

led uselessly upon the friction and other prejudicial 

be :ransmis"i 8 condly. Into that 

in the various moving elements of the machine, 

and Thirdly. Into the useful work, or that 

^e operator*, wh- . immediately the e 

products of the machine. 



THE THEORY OF MACHETES. 147 



145. The aggregate number of units of useful works yielded 
by any machine at its working points is less than the num- 
ber received upon the machine directly from the moving 
power, by the number of units expended upon the prejudi- 
cial resistances and by the number of units accumulated 
in the moving parts of the machine whilst the work is being 
done* 

For, by the principle of vis viva (Art. 129.), if sUj repre- 
sent the number of nnits of work received upon the machine 
immediately from the operation of the moving power, 2u 
the whole number of such units absorbed in overcoming the 
prejudicial resistances opposed to the working of the ma- 
chine, 2U 2 the whole useful work of the machine (or that 
done by its operators in producing the useful effect), and 

— ^w{y*—v?) one half the aggregate difference of the vires 

vivas of the various moving parts of the machine at the 
commencement and termination of the period during which 
the work is estimated, then, by the principle of vis viva 
(equation 108), 

iTJ-2TJ i + Zu+±-2wW-v:) (112) ; 

in which v 1 and v 2 represent the velocities at the commence- 
ment and termination of the period, during whichthe work 
is estimated, of that moving element of the machine whose 
weight is w. Kow one-half the aggregate difference of the 
vires vivas of the moving elements represents the work accu- 
mulated in them during the period in repect to which the 
work is estimated (Art. 130.). Therefore, &c. 



146. If the same velocity of every part of the machine n - 
turn after any period of time, or if the motion he periodical^ 
then is the whole work received upon it from themoving / 
during that time exactly equal to the sum of tfo useful work 
done, and the work expended upon the prejudicial r< sisti 
For the velocity being in this case the Bame al the com- 
mencement and expiration of the period during which the 
work is estimated, 2w(v*— 0=0, so that 

* Note (0 Ed. App. 



US mi: MODULUS OF a machine. 

2U a =2TJ,+2ti (113). 

Therefore, &c 

The convene of this proposition is evidently true. 



147. If the prime mover in a machine b> throughout tht 
motion in equilibrium with tie useful and th 
resistances^ then the '/notion of the machine is uniform. 
For in this case, bv the principle of virtual velocities 
(Art 127.), 2U 1 =2U 1 +2«; therefore (equation 112) 
'Zwiy*— v*)=0 ; whence it follows that (in the case sup- 
posed) the velocities v 1 and v 2 of any moving element of the 
machine are the same at the commencement and termi- 
nation of any period of the motion however small, or that 
the motion of every such element is a uniform motion. 
Therefore, &c. 

The converse of this proposition is evidently true. 



The Modulus of a Machine moving with a uniform or 
periodical Motion. 

148. The modulus of a machine* in the sense in which the 
term is used in this work, is the relation between the work 
constantly done upon it by the moving power, and that con- 
stantly yielded at the working points, when it has attained 
a state of uniform motion f it admit of such a state of 
motion ; or if the nature of its motion te periodical, then 
is its modulus the relation between the work done at its 
moving and at its working points in the interval of time 
which it occupies in passing from any given velocity to the 
same velocity again. 

The modulus is thus, in respect to any machine, the parti- 
cular form applicable to that machine of equation (113), and 
being dependent for its amount upon the amount of work 2 u 
expended upon the friction and other prejudicial resistances 
opposed to the motion of the various elements of the ma- 
chine, it measures in respect to each such machine the loss 
of work due to these causes, and therefore constitutes a true 
standard /'<>/• comparing th litnre of moving power ne- 

cessary to the production of the same effects by different mar 



THE MODULUS OF A MACHINE. 149 

chines : it is thus a measure of the working qualities of 
machines.* 

Whilst the particular modulus of every differently con- 
structed machine is thus different, there is nevertheless a 
general algebraical type or formula to which the moduli of 
machines are (for the most part and with certain modifica- 
tions) referable. That form is the following, 

U 1= =A . U 2 + B . S (114), 

where TJ 1 is the work done at the moving point of the ma- 
chine through the space S, U 2 the work yielded at the work- 
ing points, and A and B constants dependent for their value 
upon the construction of the machine : that is to say, upon 
the dimensions and the combinations of its parts, their 
weights, and the co-efficients of friction at their various rub- 
bing surfaces. 

It would not be difficult to establish generally this form of 
the modulus under certain assumed conditions. As the mo- 
dulus of each particular machine must however, in this work, 
be discussed and determined independently, it will be better 
to refer the reader to the particular moduli investigated in 
the following pages. He will observe that they are for the 
most part comprised under the form above assumed; sub- 
ject to certain modifications which arise out of the discus- 
sion of each individual case, and which are treated at length. 



149. There is, however, one important exception to this 
general form of the modulus : it occurs in the case of ma- 
chines, some of whose parts move immersed in fluids. It is 
only when the resistances opposed to the motion of the parts 
of the machine upon one another are, like those of friction, 
proportional to the pressures, or when they are constant re- 
sistances, that this form of the modulus obtains. If there be 
resistances which, like those of fluids in which the moving 
parts are immersed (the air, for instance), vary with the velo- 
city of the motion, and these resistances be considerable, 
then must other terms be added to the modulus. This bud- 
ject will be further discussed when the resistances of fluids 
are treated of. It may here, however, be observed, thai if 
the machine move umf&nril/y subject to the resistance of a 
fluid during a given time T, and the resistance of the fluid 

* The properties of the modulus of a machine are here, for the Aral time, 

discussed. 



THE MODCLCS OF A MACHESE. 

a raare of the velocity V, then 
will the work ex] .ce vary 

S=V . T. I:* then U. and U, represent the 
e at the moving and working points during the 
time T, then does the modolus (equation 114) assume, in this 
he form 

".=A. U,+B . V. T . T 



A MaCHTSE MOVE!»G WITH 
OR RETARDED MoTI 

. In the two last articles the work I" ;pon the 

moving point or points of the machine. an supposed to 

be just that necessary arcome the useful and prejudi- 

cial resistance sed to the motion of the machine, either 

continually or periodically ; so that all the work may be ex- 
pended upon these resistances, and none accumulated in the 
moving parts of the machine as the work proceeds, or else 

iie accumulated work may return to the same amount 
from period to period. Let us now suppose this equal H 

. and the workU : done by the moving po :eeed 

that nece— ;-ome the useful and prejudicial resist- 

ances ; and to distinguish the work represented by U, in the 
one case from that in the other, let us suppose the former 

which is in excess of the resistances ) to be represented 
by U 1 ; also let U, be the useful work of the machine, done 
through a given space S s , and which is supposed the same 
whatever may be the n of the machine 

whilst that space is being described ; moreover, let S. be the 
space described by the moving point, whilst the space 
being described by the working point. 

since F. is the work which must be done at the 
ing point just to overcome the resistances opposed to 
the motion of that point, and U 1 is the work actually done 
upon that point by tL re U— U. is the c 

of the work done by the povr -hat expended on the 

-i is therefore equal to the work accumulated 
in the machine «Art. 190.); thai one half of the 

increase of the vis viva through th S - Lrt 1 _ 

that, if r, represent the vel ^ment of the 

machine _ rhen the work TJ 1 began to be 

when that work has been com- 
-<*\ * 



THE VELOCITY OF A MACHINE. 15 J 



Now by equation (114) U 1 =AU a + BS 1 , 
.-. W=A . U 2 + B . & 1 +Yg Ew W-<) (liti> 

If instead of the work IP done by the power exceeding that 
IT, expended on the resistances it had been less than it, then, 
instead of work being accumulated continually through the 
space S l5 it would continually have been lost, and we should 
have had the relation (Art. 129.), 

so that in this case, also, 

The equation (116) applies therefore to the case of a 
retarded motion of the machine as well as to that of an 
accelerated motion, and is the general expression for the 
modulus of a machine moving with a variable motion. 
Whilst the co-efficients A and B of the modulus are depen- 
dent wholly upon the friction and other direct resistances to 
the motion of the machine, the last term of it is wholly 
independent of all these resistances, its amount being deter- 
mined solely by the velocities of the various moving ele- 
ments of the machines and their respective weights. 



The Telocity of a Machine moving with a variable 

Motion. 

151. The velocities of the different parts or elements of 
every machine are evidently connected with one another by 
certain invariable relations, capable of being expressed by 
algebraical formulae, so that, although these relations are 
different for different machines, they are the Bame for ail 
circumstances of the motion of the same machine. La a 
great number of machines this relation is expressed by a 
constant ratio. Let the constant ratio of the velocity y, of 
any element to that V, of the moving point in such a 



152 THE TELOCITY OF A MACHINE. 

machine be represented byX, so that v 1 =xY i) and .et v a and 
\\ be anj other values of v x and V, ; then s =XV r Sub- 
Btitnting these values of i^ and v^ in equation (116), we 
have 

U'=A . U,+B . 8 l+ ^y;-Y&w\' (117); 

in. which expression 2?/A 3 represents the sum of the weights 
of all the moving elements of the machine, each being mul- 
tiplied by the square of the ratio X of its velocity to that of 
the point where the machine receives the operation of its 
moving power. For the same machine this co-efficient 2io>? 
is therefore a constant quantity. For different machines it 
is different. It is wholly independent of the useful or pre- 
judicial resistances opposed to the motion of the machine, 
and has its value determined solely by the weights and 
dimensions of the moving masses, and the manner in which 
they are connected with one another in the machine. 
Transforming this equation and reducing, we have 

T>T.- +% j!£=*jg F i&| ( „„ 



by which equation the velocity Y 3 of the moving point of 
the machine is determined, after a given amount of work 
U l has been done upon it by the moving power, and a given 
amount U 2 expended on the useful resistances ; the velocity 
of the moving point, when this work began to be done 
being given and represented by Y r 

It is evident that the motion of the machine is more 
equable as the quantity represented by 2wX a is greater. 
This quantity, which is the same for the same machine and 
different for different machines, and which distinguishes 
machines from one another in respect to the steadiness of 
their motion, independently of all considerations arising out 
<>t' the nature of the resistances useful or prejudicial opposed 
to it. may with propriety be called the co-efficient of 
equable motion.'' The actual motion of the machine is 
more equable as this co-efficient and as the co-efficients A 
and B (supposed positive) are greater. 

* The co-efficient of equable motion is here, for the first time, introduced 
into the consideration of the theory of machines. 



CO-EFFICIENTS OF THE MODULUS. 153 



To determent; the Co-efficients of the Modulus of a 

Machese. 

152. Let that relation first be determined between the 
moving pressure T 1 upon the machine and its working pres- 
sure P 2 , which obtains in the state bordering upon motion by 
the preponderance of P r This relation will, in all cases 
where the constant resistances to the motion of the machine 
independently of P 2 are small as compared with P 2 , be 
found to be represented by formulae of which the following 
is the general type or form : — 

?,=?,. * t +*, (119); 

where * x and $ 2 represent certain functions of the friction 
and other prejudicial resistances in the machine, of which 
the latter disappears when the resistances vanish and the 
former does not; so that if */ 0) and * a (°) represent the 
values of these functions when the prejudicial resistances 
vanish, then * 9 (°) = and * 1 (°):= a given finite quantity 
dependent for its amount on the composition of the machine. 
Let PjW represent that value of the pressure P x which would 
be in equilibrium with the given pressure P 2 , if there were 
no prejudicial resistances opposed to the motion of the 
machine. Then, by the last equation, P^^P, . *,< >. 

But by the principle of virtual velocities (Art. 127.), if 
we suppose the motion of the machine to be uniform, so 
that Pj and P 2 are constantly in equilibrium upon it, and if 
we represent by S x any space described by the point of 
application of P„ or the projection of that space on the 
direction of P, (Art. 52.), and by S 2 the corresponding 
space or projection of the space described by P 2 , then 
P x (°) . S^P, . S 2 . Therefore, dividing this equation by 
the last, we have 

S.=§r, (120). 

Multiplying this equation by equation (119), 



•S,— P a .S 2 . f 



(+s,{^}=p,.8,{^(+s..*.; 



\A^ 



l.")4r 



which is the modulus of the machine, so that the constant 
A in equation (114) is represented by t-^, and the constant 

13 by *,. 

The above equation has been proved for any value of S„ 

ided the values of P. and P, be constant, and the 

motion of the machine uniform; it evidently obtains, there- 

. for an exceedingly small value of S x , when the motion 

of the machine is variable. 



General Condition of the State bordering- upon Motion 
en* a Body acted upon by Pressures in the save Plane, 
and moveable about a cylindrical axis. 



153. If any number of pressures P : , P„. P„ dkc. applied in 
the same plane to a body moveable about a cyUnd 

. in the state bordering upon motion, then is the 
direction of the resistance of the a.eis incU 
at the jfomt vol iterseets the circumf- . erf an 

angle equal to the limiting angle of r \ 

For let R represent the resultant of P, P,. Arc. Then, 
since these forces are supposed to be upon the 
point of causing the axis oi the bodv to turn 
upon its bearings, their resultant would, if made 
to replace them, be also on the point oi causing 
the axis to turn on its bearings. Hence it fof 
lows that the direction of this resultant R cannot 
be through the centre C of the axis : for if it 
7 were, then the axis would be pressed by it in the 
direction of a radios, that is. \z/*ty 

upon its bearings, and could not be made to turn upon them 
by that pressure, or to be upon the point of turning upon 
them. The direction of R must then be on one side of C, 
38 the axis upon its bearings in a direction RL, 
inclined to the normal CL (at the point L. where it inter- 
sects the circumference of the axis) at a certain angle RLC. 
Moreover, it is evident <Art. 141.1. that since this force R 
'ng the axis upon its bearings at L is upon the point of 
causing it to dip npon them, this inclination RLC of R to 
the perpendicular CL is equal to the limiting angle of 








THE WHEEL A2FD AXLE. 155 

resistance of the axis and its bearings.* Xow the resistance 
of the axis is evidently equal and opposite to the resultant 
E of all the forces P x , P 2 , &c. impressed upon the body. 
This resistance acts, therefore, in the direction LE, and is 
inclined to CL at an angle equal to the limiting angle of 
resistance. Therefore, (fee. 



The Wheel and Axle. 

154. The pressures P, and P 2 applied ver- 
tically by means of parallel cords to a 
wheel and axle are in the state bordering 
upon motion by the preponderance of P l5 
it is required to determine a relation 
between JP, and P 2 . 

The direction LE of the resistance of the axis is on that 
side of the centre which is towards P 1? and is inclined to the 
perpendicular CL at the point L, where it intersects the 
axis at an angle CLE equal to the limiting angle of resist- 
ance. Let this angle be represented by <p, and the radius 
CL of the axis by p ; also the radius CA of the wheel by a„ 
and that CB of she axle by a 2 ; and let W be the weight of 
the wheel and axle, whose centre of gravity is supposed to 
be C. ISTow, the pressures P n P 2 , the weight W of the 
wheel and axle, and the resistance E of the axis, are pres- 
sures in equilibrium. Therefore, by the principle of the 
equality of moments (Art. 7.), neglecting the rigidity of the 
cord, and observing that the weight W may be supposed to 
act through C, we have, 

P>. CA=P 2 . CB + E. Cm. 

If, instead of P x preponderating, it had been on the point 
of yielding, or P 2 had been in the act of preponderating, 
then E would have fallen on the other side of C, and we 
should have obtained the relation P, . CA=P 2 . CB— 
E . C^; so that, generally, P, . CA = P 2 . CB±E .Cm; 
the sign ± being taken according as P, is in the superior <>r 
inferior state bordering upon motion. 

Now CA=a„ CB=# 2 , Cm — QL sin. CLE=p sin. $, and 

* The side of C on which RL falls is manifestly determined by the direction 

towards which the motion is about to take place. In this case it is supposed 
about to ta.kc place to the right of C. If if had been u> toe ■'■/■'. th< 
tion of R would have been on the opposite side of 0. 



156 THI AND AXLE. 

B -P, + P,± W; the sign ± being taken according as the 
jht W of the wheel and axle acts in the same direction 
with the preeenree P, and P„ or in the opposite direction; 
that : 1 i ii -a as the pressures P, and P, act vertically 

dmo , r \rds (as anown in the figure) or upwards ; 

.•.PA=PA+(P.+P,±W) P BiiL 

.\ l\(a 1 — p sin. <p)=P,(a, + p sin. v)±W p sin. <p. 

Now the effect (Art. 142.) of the rigidity of the cord BP, 
is the Bame as though it increased the tension upon that cord 

(T) -i~"F P \ 
P,+ '- — *): allowing, therefore, for the 
a t ' 

rigidity of the cord, we have finally 

(T\ ITT "P \ 
P a -h ^ J_J j ( fl « + P Sin ' °)± W P 8in « <P> 

or reducing, 

P- P Lgte! + U / . . (122), 

1 5 \ aja l — p sin. <j> #,— p sin. <p v /7 

which is the required relation between P l and P, in the 
state bordering upon motion. 

p p 

— sin. <p and — sin. p are in all cases exceedingly 6mall ; 

we may therefore omit, without materially affecting the 
result, all terms involving powers of these quantities above 
the first, we .-hall thus obtain by reduction 



155. The modulus of uniform motion in the wheel and axle. 

It is evident from equation (122), that, in the case of the 
wheel and axle, the relation assumed in equation (1VJ) 

, . .« i / . E\a,-hp sin. <p 
obtains, if we take *. = l 1 +— I = ; 

1 \ aJa t — o sin. a> ' 



THE WHEEL AND AXLE. 



157 



D + (— ±WWn.<p 

and *>,= ^ J. , 

a x — p sin. 9 

Now observing that &M represents the value of *, when 
the prejudicial resistances vanish (or when 9=0 and E=0), 

we have * (°>=— . 

M *i (0) \ aja\a — psin.9/ \ aj /j>_\ sin. 9 ' 
Therefore by equation (121), 

1 + (£) sin. „ 



D-f- I— ±w\ p sin. 9 



a l — p sui. 9 



• • • (124), 



which is the modulus of the wheel and axle. 

p 
Omitting terms involving dimensions of — sin. 9, and 

a l 

p . E 

— sin. 9, and — above the first, we have 

, 1 1 W\ ) 



156. The modulus of variable motion in the wheel and aaik . 

If the relation of P, and P, be not that of either Btate 
bordering upon motion, then the motion will be continual] y 
accelerated or continually retarded, and work will continu- 
ally accumulate in the moving parte of the machine, or the 
work already accumulated there will continually expend 



158 



THE WHEEL AND AXLE. 



itself until the whole is exhausted, and the machine is 
brought to rest. The general expression for the modulus in 
this state of variable motion is (equation 116) 



!Now in this case of the wheel and axle, if V, and Y 2 re- 
present the velocities of P, at the 
■ commencement and completion 
of the space S„ and a the angular 
- velocity of the revolution of the 
| wheel and axle ; if, moreover, the 
J pressures P, and P 2 be supposed 
to be supplied by weights sus- 
pended from the cords ; then, 
since the velocity of P 2 is repre- 

a Y 

sented by - J — -, we have 2wv*=. 




l Y 1 2 +P 2 (^ 1 J+aV 1 I 1 +«V a I,,if 



I x represent the moment of inertia of the revolving wheel, 
and I 2 that of the revolving axle, (Art. 75.), and if f\ repre- 
sent the weight of a unit of the wheel and |* 2 of the axle ; 
since 2wv? represents the sum of the weights of all the mov- 
ing elements of the machine, each being multiplied by the 
square of its velocity, and that (by Art. 75.) aVJ, represents 
this sum in respect to the wheel, and a> 2 I, in respect to the 
axle. Now, V 1 =aa 1 , 



-*,.-.* V-h^^ + ^x 3 



Y, 
a, 



p.* 

«,- a," 

P,a 1 ' + PX' + «,I 1 +fSl, 



Y , 1 1 P,«,'+Pa'+i*,I.+h,I. I _ 
Similarly *<=V,' ) *A' + Prf+i>,I,+*J, j . 



Substituting in the general expression (equation 116), we 
have 



THE WHEEL AND AXLE. log 



tj i =au 2 +bs 1 +1(y;-y 1 9 ) 






j . . . (126), 



which is the modulus of the machine in the state of variable 
motion, the co-efficients A and B being those already deter- 
mined (equation 124), whilst the co-efficient 

— i-i ?_L_ — lj i_i is the co-efficient 2wX 2 (equation 

117) of equable motion. If the wheel and axle be each of them 
a solid cylinder, and the thickness of the wheel be & 1? and the 
length of the axle 5 2 , then (Art. 85.) I 1 =J*5 1 a 4 1 , I 3 =J^5 9 a 2 *. 
Now if Vj and W 2 represent the weights of the wheel and 
axle respectively, then W l =^a 1 2 b 1 ^ "W \=«a*b 2 iJ. 2 ; therefore 
f* 1 I 1 =iW 1 a 1 a , M' 9 I 9 =iW a a a 3 . Therefore the co-efficient of 
equable motion is represented by the equation 

^ v= PA , +PA'+i^ r A , +tW a fl,' or 
a* ' 

2^=P i +iW 1 + (P 2 +^)(|- 2 ) 2 (127). 



157. To determine the velocity acquired through a given 
space when the relation of the weights P x and P 2 , suspended 
from a wheel and axle, is not that of the state bordering 
upon motion* 

Let S, be the space through which the weight P, moves 
whilst its velocity passes from V l to Y 2 : observing that 

U^P.S,, and that U 2 =P a S a =P a ^- 2 , substituting in equa- 

a x 

tion (126), and solving it in respect to V„, we have 

* Note (w) Ed. App. 



160 



T1IE PULLET. 



making tlie same suppositions as in formula 127, and repre- 
senting the ratio — by m, we have 

V=V*+&iS \ P.-A . P,ot- B ) 

». > . -r -yo, i (Pi +i w i )+(P.+iW>' J • 



The Pulley. 

15S. If the radius of the axle be taken equal to that of the 
wheel, the wheel and axle becomes a pul- 
ley. Assuming then in equation 122, 
a i =a i =a, we obtain for the relation of the 
moving pressures P, and P 2 , in the state 
bordering upon motion in the pulley, when 
the strings are parallel, 




'•=■■■(>+?) 



l+-sin.p 
a 

1— -sin.; 
a 



+ 



D + (?±w) P sin.p 



a— p sin. 9 



.(129); 



and by equation 124 for the value of the modulus, 



1-f-sin. <p 
a 




(130) ; 



in which the sign ± is to be taken according as the pressures 
P, and P, act downwards, as in the first pulley of the pre- 
ceding figure ; or upwards, as in the second. Omitting 

■p 

dimension of - sin c, - sin. <p, and — above the first, we have 
a a a 

by equations (123, 125) 






SYSTEM OF ONE FIXED ONE MOVEABLE PULLEY. 



161 



Also observing that a 1 =a i , and I,=0, the modulus of varia- 
ble motion (equation 126) becomes 

IP^AU.+M+^V^Y.OiP.+P.+iWJ (133), 

and the velocity of variable motion (equations 118, 128) is 
determined by the equation 

V=Y,- + ^{ g;;^- B } (134); 

in which two last equations the values of A and B are those 
of the modulus of equable motion (equation 125). 



System of one fixed and one moveable Pulley. 

159. In the last article (equation 131) it was 
shown that the relation between the tensions 
P, and P 2 upon the two parts of a string pass- 
ing over a pulley and parallel to one another, 
was, in the state bordering upon motion by the 
preponderance of P>, represented by an expres- 
sion of the form Y 1 =aF i + b, where a and b are 
constants dependent upon the dimensions of the 
pulley and its axis, its weight, and the rigidity 
of the cord, and determined in terms of these 
elements by equation 131 ; and in which ex- 
pression b has a different value according as the 
tension upon the cord passing over any pulley 
acts in the same direction with the weight of that pulley | as 
in the first pulley of the system shown in the figure), or in 
the opposite direction (asm the second pulley): let these 
different values of b be represented by b and h v Now it is 
evident that before the weight P 2 can be raised by means of 
a system such as that shown in the figure, composed of one 
fixed and one moveable pulley, the state of the equilibrium 
of both pulleys must be that bordering upon motion, which 
is described in the preceding article; since both must be 
upon the point of turning upon their axes before the weight 
P, can begin to be raised. If then T and t represenl the 
tensions upon the two parts of the string which pass round 
the moveable pulley, we have 




162 SYSTEM OF ONE SIZED AND ANY 

l\=aT + b, andT=o*+& l . 

Now the tensions T and t together support the weight P„ 
and also the weight of the moveable pulley, 

Adding al to both sides of the second of the above equa- 
tions, and multiplying both sides by a, we have 

a(l + a)T=a*<? + t) + al l =aX? t + W) + ab l . 

Also multiplying the first equation by (1 + a), 

(1 + a)P, = a(l + a)T + 5(1 +*)= a\T, + W) + ^ + 6(1 + a), 

Now if there were no friction or rigidity, a would evi- 
dently become 1 (see equation 121), and <!>'= 1 would 

become-; the co-efficients of the modulus (Art. 148.) are 

therefore A^d£-\ and B= ^ + f + «) + °\ 
\l + a/ 1 + a 

. tt_o/ «' \tt i «'W+S(l+a)+a5 l c1 

which is the modulus of uniform motion to the single move- 
able pulley.* 

If this system of two pulleys had been 
arranged thus, with a different string passing 
over each, instead of with a single string, as 
shown in the preceding figure, then, represent- 
ing by t the tension upon the second part of 
the string to which P, is attached, and by T 
that upon the first part of the string to which 
P 2 is attached, we have 

T,=at + b, T=aP, + b y P 1 -h^W=T. 

* The modulus may be determined directly from equation (135); for it is 
evident that if S! and S 3 represent the spaces described in the same time by 
P! and P 2 , then Si = 2S 2 . Multiplying both sides of equation (13?) by thia 
equation, we have, 

now PiSi=U 1 , and P2S2 = U 2 , therefore &c. 




NUMBER OF MOVEABLE PULLEYS. 



163 



Multiplying the last of these equations by a, and adding it 
to the first, we have T 1 {l + a) + Wa=Ta+b==a r P t i-(l-\-a)bi 

and for the modulus (equation 121), 

Mi^M^SK cu* 

It is evident that, since the co-efficient of the second term 
of the modulus of this systen is less than that of the first 
system (equation 136) (the quantities a and b being essen- 
tially positive), a given amount of works U 2 may be done by 
a less expense of power U a , or a gived weight P 2 may be 
raised to a given height with less work, by means of this 
system than the other ; an advantage which is not due 
entirely to the circumstance that the weight of the move- 
able pulley in this case acts in favour of the power, whereas 
in the other it acts against it ; and which advantage would 
exist* in a less degree, were the pulleys without weight. 



A System of one fixed and any Number of moveable 

Pulleys. 



160. Let there be a system of n moveable 
pulleys and one fixed pulley combined as 
shown in the figure, a separate string passing 
over each moveable pulley ; and let the ten- 
sions on the two parts of the striDg which 
passes over the first moveable pulley be re- 
presented by T/and t t , those upon the two 
parts of the string which passes over the 
second by T, and £,, &c. Also, to simplify 
the calculation, let all the pulleys be sup- 
posed of equal dimensions and weights, and 
the cords of equal rigidity ; 




T^o^+fc,, and T a +W=T\-K ; 



eliminating, T i=(x^) T ! 



+ 



Wa + b, 
1+a 



(139). 



Let the co-efficients of this equation be represented by a 
and /3; 



164 



SYSTEM OF ONE FIXED Ax\D ANY 



Similarly, T 2 =aT, + /3, T 3 =aT 4 + /3,. T 4 =aT 6 + /3, &c. = &c 

Multiplying these equations successively, beginning from 
the second, by a, a 3 , a 8 , &c., a 71 - 1 , adding them together, and 
striking out terms common to both sides of the resulting 
equation, we have 

T 1 =a«P 3 + /3 + a/3 + a 3 /3 + .... +a— 1/3; 

or summing the geometrical progression in the second 
member, 

T 1= «»P,+/3(^) (140); 

Substituting for a and /3 their values from equation (139), 
and reducing 

NowP i =aT 1 +5; 

••• p -= a (r^) np -+«( w «+ J .)ji-(ifJ"}+s---a«)- 

Whence observing, that, were there no friction, a would 

(Cb \ n [\\ n 
- J = I J . We have (equation 121) 

for the modulus of this system, 



161. If each cord, instead of having one of 
its extremities attached to a fixed obstacle, had 
been connected by one extremity to a move- 
able bar carrying the weight P 2 to be raised 
(an arrangement which is shown in the second 
figure), then, adopting the same notation as 
before, we have 

Adding these equations together, striking out 
terms common to both sides, and solving in 
respect to T„ we have 




<.=fe)<.-fcliK 



NUMBER OF MOVEABLE PULLETS. 165 

in which equation it is to be observed, that the symbol b 
does not appear ; that element of the resistance (which is 
constant), affecting the tensions t x and £ 2 equally, and there 

fore eliminating with T x and T 2 . Let zr be represented 

by a ? then 

t x =ab — -W. Similarly, t t =at t — -W, 

* * I :, ... . , (143). 

Eliminating between these equations precisely as between 
the similar equations in the preceding case (equation 140), 
observing only tha,t here /3 is represented by — JW, and that 
the equations (143) are n— 1 in number intead of n, we have 



t = OL n -H 



af/a»-l- 1 



a 



F=?k- •■•<-) 



Also adding the preceding equations (143) together, we have 

t x + t,+ . . . +*.-! = «(*, + *.+ . . . 0-(^-l)— • 

<Z 

Now the pressure P 2 is sustained by the tensions tf„ £ 2 , &c. 
of the different strings attached to the bar which carries it. 
Including P 2 , therefore, the weight of the bar, we have 

#,+*,+ . . . + * n _i+* n =P 2 ; .■.*+*+ • • +#.-i=P.-*.5 
and£ 3 + . . . +£ j =P a — ^; 

0/ 

Substituting this value of t n in equation (144), 

,_ (1 _ aK - iP]+a%+(re _ 1) ^_«w/ a _ r ^i\ 

v J a a \ « — 1 / 

Transposing and reducing, 

(i-«^ 1 =(i-«)«- 1 P,+- j^-^-j^J ; 

"*""" l-a» 9+ a (1-a* 1-*)' 



1G6 TACKLE OF ANY CT7HBEB OF SHEAVES. 

(1— aW*- 1 a-* — 1 
-a, -, aiiu 



Now a 



, . , .a- 1 =l+«" 1 ; also 



a-' 1 — 1 



and 



«; 



(l + a- 1 /*— 1 1— a* tt-*—l (1 + ar 1 / 1 — 1 1— a 

q-*P, WC n 1 

" ' (l + a-y-l a \\jL + ar x ) n —\ ) 
Xow P^a^ + 5; 

^WTience observing that when 0=1, [(l + a~~ l j n — 1\ =2*— \ t 
we obtain for the modulus of uniform motion (equation 
121), 



(1 + a-y-l 

| w )aT^i-4 +5 ( s "-- (U6) - 



A Tacexe of axt Xumbee of Sheaves. 

162. If an number of pulleys (called in this case sheaves) 
be made to turn on as many different centres in the same 
block A, and if in another block B there be simi- 
larly placed as many others, the diameter of each 
of the last being one hah' that of a correspond- 
ing pulley or sheave in the first ; and if the same 
cord attached to the first block be made to pass 
in succession over all the sheaves in the two 
blocks, as shown in the figure, it is evident that 
the parts of this cord 1, 2, 3, ike. passing between 
the two blocks, and as many in number as there 
are sheaves, will be parallel to each other, and 
will divide between them the pressure of a weight 
P, suspended from the lower block : moreover, 
that they would divide this pressure between 
them equally were it not for the friction of the 
*t sheaves upon their bearings and the rigidity of 

the rope ; so that in this case, if there were n sheaves, the 

tension upon each would be -P, ; and a pressure P, of that 



TACKLE OF ANY NUMBER OF SHEAVES. 



167 



amount applied to the extremity of the cord would be suffi- 
cient to maintain the equilibrium of the state bordering upon 
motion. Let T n T 2 , T 3 , &c. represent the actual tensions 
upon the strings in the state bordering on motion by the pre- 
ponderance of P l5 beginning from that which passes from P, 
over the largest sheaf; then 

&e.=&c. 



IV 



=a,T 1 +& 1 ,T 1 =a,T 1 + & i 



co- 



where a l9 <z 3 , &c, 5 1? b„ &c. represent certain constant 
efficients, dependent upon the dimensions of the sheaves and 
the rigidity of the rope, and determined by equation (131). 
Moreover, since the weight P 2 is supported by the parallel 
tensions of the different strings, we have 



P^T.+T.H-. 



4-T. 



It will be observed that the above equations are one more 
in number than the quantities T„ T 2 , T 3 , &c. ; the latter may 
therefore be eliminated among them, and we shall thus ob- 
tain a relation between the weight P 2 to be raised and that 
P, necessary to raise it, and from thence the 
modulus of the system. 

To simplify the calculation, and to adapt 
it to that form of the tackle which is com- 
monly in use, let us suppose another ar- 
rangement of the sheaves. Instead of their 
being of different diameters and placed all 
in the same plane, as shown in the last 
figure, let them be of equal diameter and 
placed side by side, as in the accompanying 
figure, which represents the common tackle. 
The inconvenience of this last mode of ar- 
rangement is, that the cord has to pass from 
the plane of a sheaf in one block to the plane, 
of the corresponding sheaf in the other ob- 
liquely, so that the parts of the cords be- 
tween^ the blocks are not truly parallel to 
one another, and the sum of their tensions is not truly equal 
to the weight P 2 to be raised, but somewhat greater than it. 
So long, however, as the blocks are not very near to one an- 
other, this deflection of the cord is inconsiderable, and the 
error resulting from it in the calculation may be neglected. 
Supposing the different parts of the cord between the blocks 
then to be parallel, and the diameters of all the Bheaves and 




103 *** - NL M; BHKAYBRi 

their axes to be eqoal, also neglecting the influence tf the 
jh Bheaf in increasing the friction of its axis, 

» 96 weights are in this case comparatively small, the 
tg . a t will manifestly all be equal; as also 

also P^T.+T.+T,* +T . 

Multiplying equations (147) successively (beginning from the 
second) by a, a\ a% and ^ n ~ 1 ; then adding them together, 
striking out the terms common to both sides, and summing 
the geometric series in the second member (as in equation 
140), we have 

p-^+afci 

a — 1 * 

Adding equations (147), and observing that T x -fT,+ 
.... +T n =P 3 , and that P^ + T,* .... +T_t= 
P/+P,— Tj we have 

P.+P.-T^oP.+n*. 

Eliminating T n between this equation and the last, 

a*(a-l) nho«_ h {U) 

•* ri a n -l ,n V— 1 a—1 K J 

To determine the modulus let it be observed, that, neglect- 
in o- friction and rigidity, a becomes unity ; and that for this 

value of a, — s --^—^- becomes a vanishing fraction, whose 
a n — 1 

value is determined bv a well known method to be -*. 

1\ 

Hence (Art. 152.), 

* Dividing numerator and denominator of the fraction by (a—1) it becomee 
, which evidently equals — when o=l. The modulus 






.-i + a ._(_... -f-l • n 

may readily be determined from equation (148). Let Sj and Sj represent the 
spaces described by Pi and P : in any the same time ; then, sinoe when the 
blocks are made to approach one another by the distance S g , each of the n por- 
tions of the cord intercepted between the two blocks is shortened by this dis- 



THE MODULUS OF A COMPOUND MACHINE. 169 

tt a n (a—l) TT ( noa n J ] n 

Hitherto no account litis been taken of the work expended 
in raising the rope which ascends with the ascending weight, 
The correction is, however, readily made. By Art. 60. it 
appears that the work expended in raising this rope (diffe- 
rent parts of which are raised different heights) is precisely 
the same as though the whole quantity thus raised had been 
raised at one lift through a height equal to that through 
which its centre of gravity is actually raised. Now the cord 
raised is that which may be conceived to lie between two 
positions of P 2 distant from one another by the space S 2 , so 
that its whole length is represented by nS 2 ; and if ^ repre- 
sent the weight of each foot of it, its whole weight is repre- 
sented by pnS 2 : also its centre of gravity is evidently raised 
between the first and second positions of P 2 by the distance 
|S 2 ; so that the whole work expended in raising it is repre- 
ss 2 
sented by i^nS^ or by J — -, since S X =^S 2 . Adding this 

work expended in raising the rope to that which would be 
necessary to raise the weight P 2 , if the rope were without 
weight, we obtain* 

Bl ^g! Il+ {* r _^j.H +£ . v .. ..(.»), 

which is the modulus of the tackle. 



The Modulus of a compound Machine. 

163. Let the wor~k, of a machine be transmitted from one 
to another of a series of moving elements forming a com- 
pound machine, until from the moving it reaches the working 
point of that machine. Let P be the pressure under which 
the work is done upon the moving point, or upon the first 
moving element of the machine ; P x that under which it is 

tance S 2 , it is evident that the whole length of cord intercepted between the 
two blocks is shortened by nS 2 ; but the whole of this cord must have passed 
over the first sheaf, therefore Si=wS 2 . Multiplying equation (148) by this 
equation, and observing that Ui^PiSj and U 2 =P 2 S 2 , we obtain the modulus 
as given above. 

* A correction for the weight of the rope may be similarly applied to the 
modulus of each of the other systems of pulleys. The effed of the vmghi of 
the rope in increasing the expenditure of work on the friction of the pulleys if 
neglected as unimportant to the result. 



170 MODULUS OF A COMPOUND MACHINE. 

yielded from the first to the second element of the machine; 
l* a from the Becond to the third element, &c. ; and P n the 
pressure under which it is yielded by the Lasl element upon 
the useful product, or at the working point of the machine. 
Then, since each element of the compound machine is a sim- 
ple machine, the relation between the pressures applied to 
that ^lenient when in the state bordering on motion will be 
found to present itself under the form of equation (119) 
! Art. 152), in all cases where the pressure under which the 
work upon each element is done is great as compared with 
the weight of that element (see Art. 166.). 

Representing, therefore, by & n a 3 , a z . . . &„ 5 9 , b s . . ., cer- 
tain constants, which are given in terms of the forms and 
dimensions of the several elements and the prejudicial resist- 
ances, we have 

&c.=&c, V^—aY^b^ 

Eliminating the n—1 quantities P x , P 2 , P 3 . . ., P^, between 
these n equations, we obtain an equation, of the form, 



P=0P. + & (151); 

where a=a 1 a n a s . . . a n , and 

b=a^ . . . « n _ 1 5 n + a x a,i . . •# n _2& n _ 1 + • • • + #A + &u 



(152). 



If the only prejudicial resistance to which each element is 
subjected be conceived to be friction, and the limiting angle 
of resistance in respect to each be represented by <p ; then 
considering each of the quantities «,, j n a 3 , b» as a function 
of <p, expanding each by Maclaurin's theorem into a series 
ascending by powers of that variable, and neglecting terms 
which involve powers of it above the first, we nave 

«.=«.» + (£)*>,, ».=».»+ (f)% «-*»! 

where, a(°\ b(°\ a(°\ b(°\ represent the values of « 15 b„ a 2 , 
b.„ &c, when <p==0 and l-^-) , \~T L ) ' ^ c * re P resent 

the similar values of their first differential co-efficients. 



AXES. 



171 



Let 



/<M (0) ™ /^i\ (0) 7^ « 



I* 



9= 






&c.= &c. 



fin \(°) 

9=^).*,, ^ jp== j r( o) 

Therefore a,= a<& (1 + "0, \= IP (1+&), 3 = a 9 (°> (1 +<0 
\=b^ (1 + /3 S ), &c.=&c. ; where a, ^ a 25 /3 25 & c ., each 
involving the factor <p, are exceedingly small. Substituting 
the values of a i5 a 2? &c in the expression for a, and neglect- 
ing terms which involve dimensions of a i5 a 2 , &c. above the 
first, we have 



a=a^ aJP> . . . aj® \1 + a x + a 2 + a 3 + + a n 



. (153). 



Now the co-efficient of the first term of the modulus is 
represented (equation 121) by — — , a representing the co- 

efficient of the first term of equation (119), also substituting 
the value of a from equation (153), and observing that 

a m =ajn . aj® a n <®, we have -?- = }l + a +. . .+<*„}; 

.•.tr = {l + ai + a a + a 8 4- .... +ajU fl + J.S .... (154), 
which is the modulus of a compound machine of n elements, 
U rej^resenting the work done at the moving point, U n 
that at the working point, S the space described by the 
moving point, and o a constant determined by equation 
(152). 



164. The conditions of the eqtjild3rium of any two pees 
suees p 2 and p 2 appld3d in the same plane to a bod^l 
moveable about a fixed axis of given dimensions. 



In Jig. 1. the pressure P x and P a are shown acting on oppo- 
site sides of the axis 
whose centre is (\ and 
mfig. 2. upon the Bame 
side. Lei the direc- 
"^ tion of tlic resultant 
of P, and I*, be repre- 
sented, in the first 
case, by IK, and in 
tile Becond by Rl. It 




172 AXES. 

is in the directions of these lines that the axis is, in the twc 
cases, pressed upon its bearings. Suppose the relation 
between P, and P, to be such that the body is, in both 
cases, upon the point of turning in the direction in which 
P, acts. This relation obtaining between P, and P 2 , it is 
evident that, if these pressures were replaced by their re- 
sultant, that resultant would also be upon the point of caus- 
ing the body to turn in the direction of P,. The direction 
IH of the resultant, thus acting alone upon the body, lies, 
therefore, in the first case, upon the same side of the centre 
C of the axis as P t does, and in the second case it lies upon 
the opposite side ;* and in both cases, it is inclined to the 
radius CK at the point K, where it intersects the axis at an 
angle CKE, equal to the limiting angle of resistance (see 
Art. 153.). Now r , the resistance of the axis acts evidently in 
both cases in a direction opposite to the resultant of P, and 
P 2 , and is equal to it ; let it be represented by R. Upon 
the directions of P n P 2 , and R, let fall the perpendiculars 
CA„ CA 2 , and CL, and let them be represented by a„ a„ 
and K Then, by the principle of the equality of moments, 
since P l5 P 2 , and R are pressures in equilibrium, 

.-.P^^P^+XR 

If P, had been upon the point of yielding, or P 2 on the 
point of preponderating, then R would have had its direction 
(in both cases) on the other side of C ; so that the last equa- 
tion would have become 

PA + VR=P 2 a 2 . 

According, therefore, as P, is in the superior or inferior 
state bordering upon motion, 

PA-R 2 ^=-(± X )K. 

And if we assume X to be taken with the sign + or — , ac- 
cording as P, is about to preponderate or to yield, then 
generally 

PA-P 2 « 2 =XR (155). 

^Now, since the resistance of the axis is equal to the resultant 
of P, and P 2 , if we represent the angle PJP, by «t> we have 
(Art. 13.) 

* The arrows in the figure represent, not the directions of the resultant* 
but of the resistances of the axis, which are opposite to the resultants. 

f Care must be taken to measure this angle, BO that Pi and P 2 may have 



AXES. 173 

R= vT 1 a + 2P 1 P, cos. « + P 2 2 . 

Substituting this value of R in the preceding equation, and 
squaring both sides, 

(P A -P 2 « 2 ) - X2 (^ 2 + 2P 1 P 2 cos. <-f P 3 2 ); 
transposing and dividing by P 2 2 , 

( J) V-V)-2 ( |i) (afi,+ V cos. .)= -(^-W) ; 

(p 
-J 



P i_(«A+ xa cos, i) ± y>A+*- 2 cos. i)'— fa'—V) «-^ a ) 



. P!_(«A+ X2 cos. i) zt X < |/(g 1 *+2g ; g a cos . »-f-a 2 2 )— >> a sin.V 
""' P, - ^k* 

Now let the radius CK of the axis be represented by p, 
and the limiting angle of resistance CKR by 9 ; therefore 
X=:CL=CK sin. CKR=p sin. 9. Also draw a straight line 
from A, to A 2 in both figures, and let it be represented by L ; 
:.a*—*2ia x a % cos. A X CA 2 + # 2 2 =L 2 . Now, since the angles at 
A x and A 2 are right angles, therefore the angles A,IA 2 and 
A^A, are together equal to two right angles, or A^CA^ + j 
==*; therefore A 1 CA 2 =^—i, and cos. A,CA a =— cos. 1; 
therefore l?=a*+2a 1 a 9 cos. < + # 2 2 : substituting these values 
of L 2 and X in the preceding equation, 

_(fl,fl s + p > cos. 1 sin. V) + p sin. 9 (L 2 — p 2 sin. 2 » sin. 9 <p)* 
1 ~~ (a x 2 — p 2 sin. 2 <p) 

. P 2 . . (156). 

The two roots of the above equation are given by positive 
and negative values of \ they correspond therefore (equa- 
tion 155) to the two states bordering upon motion. These 
two values of X are, moreover, given by positive and nega- 
tive values of 9 ; assuming therefore 9 to be taken positively 
or negatively, according as P, preponderates or yields, we 
may replace the ambiguous by the positive sign. The 

their directions both towards or both from the angular point I (as shown in the 
figure), and not one of them towards that point and the other from it. Tims, 
in the second figure, the inclination i of the pressures Pj and Ps ' s Q01 the 
angle A 2 IPi, but the angle P a lPi. It is of importance to observe this distinc- 
tion (see note p. 1*75.). 



1 74: AXK8. 

relation above determined between P, and P 9 evidently 
Batigfiee the conditions of equation (119). We obtain there 
tore for the modulus (equation 121) 

tt —i a \ ( a * a * + p a cos - * Bm - 3(p ) + p(-^ a ~ p a sm - a * em - **)* sm * v 

IT, . . . (157). 

If terms involving powers of J— ) sin. <p above the first be 

neglected, that quantity being in all cases exceedingly 
have 

H ©+(§)• sin - 1 p > « 

H 1+ © sin -*l u - (159) - 



small, we have 




165. To determine the resultant P of any number of pres- 
sures P„ P 2 , P 3 . . . ., in terms of those pressures, and the 
cosines of their inclinations to one another. 

Let a i5 a 2J a 35 &c. represent the inclinations 
IAC, IBC, &c. of the several pressures P„ 
P 2 , &c. to any given axis CA in the same 
plane; and let i M , i M , « 23 , &c. represent the 
€ inclinations of these pressures severally to one 
^ another. 

Now ZAIB=ZIBC-ZIAC (Euc. I. 32.); 

.'. » 13 r=a 2 — a 1? .'.COS. » 12 = COS. a x COS. a a -f gill. a x sin. a,. 

Similarly, cos. i 13 =cos. a, cos. a 3 + sin. a t sin. a 8 ; J 

COS. « 2 , = COS. a a COS. a, -f sill. « 2 sin. a 8 . 
NOW E 3 = (P, COS. «, + ?, COS. a, + P s COS. « 3 -f . . . ) a + (P, 

sin. ^-{-P, sin. a 2 + P 3 sin. a 3 + . . . ) 2 ,- (equation 9, Art. 11.). 

Squaring the two terms in the second member, adding the 
results, and observing that cos. "a^ain. \ = 1, 

K'^P^+P^ + P, 2 . . . +2P 1 P 1 (coB.« 1 coB.a s +8in.a 1 Bin.«s) 

+ 2P,P 3 (cos. ctj cos. a s -|-sin. a x sin. a 3 ) + . . . . ; 



AXES. 



175 



R a *=P 1 2 +P 2 2 +P 8 9 + u m a +2Pi p 2 cog> l-+3 p i p i cos . , a 
-f2P 3 P 3 cos.< 23 -f &c (160). 



166. The conditions of the equilibrium of three pees 

8URES, P 1? P 2 , P 3 , IN THE SAME PLANE APPLIED TO A BOOT 
MOVEABLE ABOUT A FIXED AXIS, THE DIRECTION OF ONE 01 
THEM, P 3 , PASSING THROUGH THE CENTRE OF THE AXIS, AND 
THE SYSTEM BEING IN THE STATE BORDERING UPON MOTION 



BY THE PREPONDERANCE OF P 1# 



Let l w , l„, » a3 




represent the inclinations of the directions of 
the pressures P l5 P 2 , P 3 to one 
another, a x and a 2 the perpen- 
diculars let fall from the cen- 
tre of the axis upon P x and P 2 , 
and X the perpendicular let 
fall from the same point upon 
the resultant E of P 1? P 2 , P 3 . 
Then, since It is equal and 
opposite to the resistance of 
the axis (Art. 153.), we have, 
by the principle of the equality of moments, T 1 a 1 —Y i a i = 
KR, for P 3 passes through the centre of the axis, and its 
moment about that point therefore vanishes. 

Substituting the value of Pfrorn equation (160), 

P 1 « 1 -PA=^{P 1 9 +P 3 9 +P, a +2P 1 P a cos. . M + 

2P 1 P 3 cos.« 13 + 2P 2 P 3 cos. i,,.}* 

Squaring both sides of this equation, and transposing, 

p;(^_V)_2P 1 |P 2 ^ 3 +X 9 (P 2 cos.« 12 +P 3 cos. i 18 )j = 

-P 2 V+^ 2 |p;+p;+2P 2 P 3 cos. « 23 | . 

If this quadratic equation be solved in respect to P„ and 



* In which expression it is to be understood that the inclination t^ of the 
directions of any two forces is taken on the supposition that both the forces 
act from or both act towards the point in which they intersect, 
and not one towards and the other from that point; so that in 
the case represented in the accompanying figure, the inclina- 
tion ii 2 of the two forces Pi and P 2 represented by the arrows, 
is not the angle PJP2, but the angle QIPi, since IQ and IP, arc 
directions of these two forces, both tending from their point 
of intersection ; whilst the directions of P«l and TP t are one 
of them towards that point, and the other from it. 




176 AXES. 

terms which involve powers of X above the first be omitted, 
we ahal] obtain the equation 

Vp a a (a 1 3 +2a 1 a a cos.« 12 +« a 2 )4-P, i a 1 a +2P 3 P 3 « 1 (^ cos^.+a, cos.«„) ; 

or representing (as in Art. 164.) the line which joins the 
feet of the perpendiculars, a x and & 2 by L, and the function 
a x (a 2 cos. ',3 + 0! cos. » 23 ) by M, and substituting for X its 
value p sin. <p, 

P,= (5)P,+ f-^) jP,"L , +P,V+2PJ , ^}«* . . . (161). 

Representing (as in Art. 152.) the value of P, when the 
prejudicial resistances vanish, or when 9=0, by Pi (0) , we 

have P/ 0) = ( — JP 2 . Also by the principle of virtual velo- 
cities PW . S!=P, . S r Eliminating PW between these 
equations, we have S t = I — ) S a . Multiplying equation (161) by 

this, P,S =P 2 S 2 +^-^-^SP 2 2 S 2 2 L 2 + 2P 2 P 3 S 2 3 M+lVS 2 V| + . 
Substituting U, for PJ3,, TJ 2 for P 2 S 2 , and observing that 



S .=S 8 » 



*=n, + ^{Ti,*w,p.s,M(|) 

P 3 3 S> 2 a I * (162.) 



which is the modulus of the system. 

If P, be so small as compared with P 2 that in the expan- 
sion of the binomial radical (equation 161), terms involving 

P s 
powers of -^ above the first may be neglected ; then, 

' i 

* It will be shown in the appendix, that this equation is but a particular 
case of a more general relation, embracing the conditions of the equilibrium 
of anv number of pressures applied to a body moveable about a cylindrical 
axis of given dimensions. 



AXES. 177 

which equation may be placed under the form 

Whence observing that the direction of P s being alwa^ 3 
through the centre of the axis, the point of application of 
that force does not move, so that the force P 3 does not work 
as the body is made to revolve by the preponderance of T l ; 
observing, moreover, that in this case the conditions of 
equation (119) (Art. 152.) are satisfied, we obtain for the 
modulus 



167. The conditions of the equilibrium of two pressures P x 
and P 2 applied to a body moveable about a cylindrical axis, 
taking into account the weight of the body and supposing it 
to be symmetrical about its axis. 

The body being symmetrical about its axis, its centre of 
gravity is in the centre of its axis, and its weight produces 
the same effect as though it acted continually through the 
centre of its axis. In equation (161.) let then P 3 be taken to 
represent the weight W of the body, and »,„ i M the inclina- 
tions of the pressures P x and P 2 to the vertical. Then 

P-g?)p,+ j^-r?-) {P/L 2 + 2P,WM+WV }*. . (165.) 

Also by the equation (162) we find for the modulus 

\ a,a, f [ a, 

+ WSX | * • • • ( 166 -) 

And in the case in which P, is considerable as compared 
with W, by equations (163, 164). 

12 



17$ 



THE DIRECTION OF THE 






168. A MACHINE TO WHICH ARE APPLIED ANY TWO PRESSURE8 
P, AND P a , AND WHICH 18 MOVEABLE ABOUT A CYLINDRICAL 
AXIS, 18 WORKED WITH THE GREATEST ECONOMY OF POWER 
WHEN THE DLRECTION8 OF THE PRESSURES ARE PARALLEL, 
AND WHEN THEY ARE APPLIED ON THE SAME SDDE OF THE 
AXIS, IF THE WEIGHT OF THE MACHLNE ITSELF BE SO SMALL 
THAT ITS INFLUENCE IN LNCREASLNG THE FRICTION MAY BE 
NEGLECTED. 

For, representing the weight of such a machine by W, and 
neglecting terms involving W sin. <p, it appears by equation 
(168) that the modulus is 



U,=U. 



1 + J — sin. 9 



a,a n 



whence it follows that the work U,, which must be done at 

the moving point to yield a given amount U 2 at the working 

point, is less as L is less. 

Now L represents 
the distance AjA, be- 
tween the feet of the 
perpendiculars CAj and 
OA„ which distance is 
evidently least when P, 
and P Q act on the same 
side of the axis, as in 
Jig. 2, and when CA, 

and CA 2 are in the same straight line ; that is, when P x and 

P 3 are parallel. 




169. A MACHINE TO WHICH ARE APPLIED TWO GTVEN PRES- 
SURES Pj AND P 2 AND WHICH 18 MOVEABLE ABOUT A CYLLN- 
DRICAL AXIS, IS WORKED WITH THE GREATEST ECONOMY OF 
POWER. THE INFLUENCE OF THE WEIGHT OF THE MACHINE 



GREATEST ECONOMY OF POWER. 179 

ARE APPLIED ON THE SAME SIDE OF THE AXIS, AND WHEN 
THE DIRECTION OF THE MOVING PRESSURE P x IS INCLINED TO 
THE VERTICAL AT A CERTAIN ANGLE WHICH MAT BE DETER- 
MINED. 

Let P 3 be taken to represent the weight of the machine, 
and let its centre of gravity coincide with the centre of its 
axis, then is its modulus represented (equation 166.) by 

U,=U,+ P -™^- 1 WL'+STT^PAmN +P s 'S 1 V I \ 

in which expression the work TJ X , which must be done at the 
moving point to yield a given amount U 2 of work at the 
working point, is shown to be greater than that which must 
have been done upon the machine to yield the same amount 
of work if there had been no friction by the quantity 

Pi!?_iliu 2 2 L 2 + 2U a P3S 1 M(~ 2 +P 3 9 S 1 vl i 
a,a t \ \a x ) 

The machine is worked then with the greatest economy of 
power to yield a given amount of work, U 2 , when this func- 
tion is a minimum. Substituting for L 2 its value 

a x + 2a x a 2 cos. » 12 + # 2 2 , and for M its value a x {# 2 cos. « 18 -f 
d^cos. < 23 | (see Art. 166.), also for Sj— j its value S 9 . it be- 



comes 
p sin 
a x a^ 



imjp ( Tj^^» H _2« 1 a 2 cos.i 12 + + 2U 2 P 3 S 2 « 1 (^cos.» 18 + 
a k coB.i M )+P i i Sx|* (169.) 



JS"ow let us suppose that the perpendicular distance a 2 from 
the centre of the axis at which the work is done, and the in- 
clination » 23 of its direction to the vertical, are both given, as 
also the space S 2 through which it is done, so that the work 
is given in every respect ; let also the perpendicular distance 
a x at which the power is applied, and, therefore, the space S, 
though which it is done, be given ; and let it be required to 
determine that inclination i M of the power to the work which 
will under these circumstances give to the above function its 
minimum value, and which is therefore consistent with the 
most economical working of the machine. 

Collecting all the terms in the function (169.) which con- 




180 TIIE DIRECTION OF THE 

tain (on the above suppositions) only constant quantities, and 
i epresenting their sum by C, it becomes 

tIH^± \ 2fl 1 a 1 U,(U 1 cos. « 12 + P 3 S 2 cos. « 18 ) + C } * 

Now C being essentially positive, this quantity is a mini- 
mum when 2# 1 <t 2 U 2 (U 2 cos. i 12 + P 3 S 2 cos. » 13 is a minimum ; or, 
observing that U 2 =P 2 S 2 and dividing by the constant factor 
2a 1 a a U 2 S 2 , when 

P a cos. » 12 + P 3 cos. i 19 is a minimum. 

From the centre of the axis C let lines Cp x 
Q? 2 be drawn parallel to the directions of the 
pressures P,P 2 respectively ; and whilst Cjp a 
and Cp, retain their positions, let the angle 
j^CPg or i 13 be conceived to increase until Y t 
attains a position in which the condition 
P 2 cos. » 12 -fP 3 cos. < 13 =a minimum is satisfied. 
Now i? I CP,=/? 1 Cp > — ^ 9 CP„ or i 13 =« 12 -« 58 ; 
substituting which value of i„ this condition 
becomes 

P 2 cos. » 12 + P 3 cos.(» 13 — » 23 ) a minimum, 

or P 2 cos. « 12 + P 3 cos. i M cos. < 23 +P 3 sin. i M sin. i 23 a minimum, 

or (P 2 4-P 3 cos. i M ) cos. » 12 + P 3 sin. i 23 sin. » ia a minimum. 

T P 3 sin. i M 

Let now^ — -^ — = tan. 7, 

P 2 + P 3 COS.« 23 

.-. (P 2 + P 3 cos. t„) cos. « 12 + (P a + P 3 cos. i„) tan. 7 sin. i H is a mi- 
nimum, or dividing by the constant quantity (P 2 + P 3 cos. i„) 
and multiplying by cos. 7, 

cos. i M cos. 7 + sin. i u sin. 7=cos. (i 13 — 7) is a minimum. 

.•, M= , +fam .- 1 { Pt ^,J ...(lY0. ] 

To satisfy the condition of a minimum, the angle jpS^JP. 
must therefore be increased until it exceeds 180° by that 

angle 7, whose tangent is represented ty p , 3 p 2 3 • To 

determine the actual direction of P, produce thenpfi to q, 
make the angle qCr equal to 7 ; and draw Cm perpendicular 



GREATEST ECONOMY OF POWER. 181 

to O, and equal to the given perpendicular distance a 1 of 
the direction of P x from the centre of the axis. If mP x be 
then drawn through the point m parallel to O, it will be in 
the required direction of P 1 ; so that being applied in this 
direction, the moving pressure P x will work the machine with 
a greater economy of power than when applied in any other 
direction round the axis. 

It is evident that since the value of the angle i M or pfip iy 
which signifies the condition of the greatest economy of 
power, or of the least resistance, is essentially greater than 
two right angles, P x and P 2 must, to satisfy that condition, 

BOTH BE APPLIED ON THE SAME SIDE OF THE AXIS. It is then 

a condition necessary to the most economical working of any 
machine {whatever may he its weight) which is moveable about 
a cylindrical axis under two given pyressures, that the mov- 
ing PRESSURE SHOULD BE APPLLED ON THAT SIDE OF THE AXIS 
OF THE MACHLNE ON WHICH THE RESISTANCE IS OVERCOME, OR 

the work done. It is a further condition of the greatest 
economy of power in such a machine, that the direction in 
tchich the moving pressure is applied should be inclined to 
the vertical at an angle i 19 , whose tangent is determined by 
equation (170.). 

When ' 23 =0, or when the work is done in a vertical 
direction, tan. y=0; therefore *„=*, whence it follows that 
the moving power also must in this case be applied in a ver- 
tical direction and on the same side of the axis as the work. 

When « 23 =^ or when the work is done horizontally, tan. 



p. 



».„=* + tan. 



©■ 



The moving power must, therefore, in this case, be applied 
on the same side of the axis as the work, and at an incli- 
nation to the horizon whose tangent equals the fraction 
obtained by dividing the weight of the machine by the 
working pressure. 

3* 
Since the angle » 12 is greater than * and less than -=■ 

cos. »„ is negative; and, for a like reason, cos. »„ is also in 
certain cases negative. Whence it La apparent that the 
function (169.) admits of a minimum value under certain 



183 



THE PULLEY. 



not only in respect to th < of the 

movu _ -are. but in a, of its 

direction from the centre of the axis. It" we suppose the 

- through which the power acts whilst the g 
amount of work U. is done to bv . and substitute in 

that function for the product S - B . and then 

B8G me the differential of the function in : to a 1 to 

vanish, we shall obtain bv reduction 



a 1 = —a i . 



U,'^2U,P,S 1 c_qs...,,^P, 1 S,' 
0," cos. i„+ U^P.S, cos. i„ 



(in.) 



If we proceed in like manner assuming the space S, instead 
of S. to be constant and substituting "in the function . 
S,fl^ its value S./.Z.. we shall obtain by reduction 



■':■ 



Pa 



P, cos. « 15 -rP, cos. i ls . 



It is easily seen that if when the values of » 12 and «„ deter- 
mined bv equation (170.) are substituted in these equations, 
the resulting values oi >.i, are . they c nd in 

the two cases to minimum values oi the functi . and 

determine completely the conditions of the greal - oomy 

of power in the machine, in respect to the direction of the 
moving pressure applied to it. 



17". The pullet, when- the tensions upon the two 
extremities of the cord have not vertical direc"- 




In the case in which the two parts oi the 

string which pass alley are not 

parallel to one another, the relat: - 

fished in Article 158. no longer obtain; 

and we must have recourse to equation 

to establish a relation between the 

ms upon them in th lering 

■ion. Calling "W the weight of 

the pulley, a irs radius, and observing 

that the effect of the rigidity of the 

in increasing the tension P.. is the same 

as though it caused the tension P, to be- 

ED 

come P. ( 1 ) iArt. 142A we have 

M a] a 



THE PULLEY. 183 

j p ( H , E\ . D i ,M P . 

Ei 



.•.p,=(i+!)ji + ^»|p. + 



D DL . IW 

-+-^rp 6111.9+-^ .psm.9; 



or, 



(?+iisi)'« to -»(- v ci«.) 

where L represents the chord AB of the arc embraced by 
the string, and M=<z 2 (cos. » ls + cos. » 23 ), < 13 and i„ represent- 
ing the inclinations of P x and P 2 to the vertical: which 
inclinations are measured by the angles P^P,, and P 2 FP 3 . 
or their supplements, according as the corresponding pres- 
sures P a and P 2 act downwards, as shown in the figure, or 
upwards (see note to Article 165.) ; so that if both these 
pressures act upwards: then the cosines of both the angles 
become negative, and the value of M becomes negative ; 
whilst if one only acts upwards, then one term only of the 
value of M becomes negative. 

Substituting this value for M, observing that Jj=2a cos. », 
where 2j represents the inclination of the two parts of the 
cord to one another (so that 2i=i w +* a3 ), and omitting terms 
which involve products of two of the exceedingly small 

.. • £> E 3 P - 

quantities — , — , and -sin. 9 we have 

^ a a 9 a 



E 2p . ) ^ D TVp(cos. i 13 + cos. i M )sin.? 

— +-cos.«sm.9 h p 2 +-+ — o^T^ — 

a a J a 2a cos. < 

1 E 2p ) ( D 

u H 1+ W cos - ,sin -*! u >M" + 

W P (cos. » 13 + cos.i a8 )sin.9 ) g # 

2a cos. « ) ' ' ' )y 



1S4 



THE PULLEY. 



which last equation is the modulus to the pulley, when the 

two parts of the string are inclined to the vertical and to 
one another. 

171. If both the strings be inclined at equal angles to the 
vertical, on opposite sides of it; or if i 13 =» 23 =i ? so that cos. 
i n +cos. « 23 =2 cos. »j then equations (172.) and (173.) become 

p »= { 1+ a + i cos -' sm - 9 [ p .+7+T-"-*- • • (1T4 )> 



H 



1-f — +— cos. i sin. 9 
a a 



172. If both parts of the cord passing over a pulley be in 
the same horizontal straight line, so that the 
pulley sustains no pressure resulting from the 
tension upon the cord, but only bears its 




weight, then i==, and the term involving 

cos. i in each of the above equations vanishes. It is, how- 
ever, to be observed that the weight bearing upon the axis 
of the pulley is in this case the weight of the pulley 
increased by the weight of cord which it is made to support. 
So that if the length of cord supported by the pulley be 
represented by », and the weight of each foot of cord by y-, 
then is the weight sustained by the axis of the pulley repre- 
sented by W-f /AS. Substituting this value for AY in equa- 
tion (175.), and assuming cos. i=0, we have 

!T.=(l+f)n+5 | D + (W+m*) P sin. 9 | S, . . . . (176.) 



173. 



Let us now suppose that there are n equal pulleys 
sustaining each the same length 
s of cord, and let U„ represent 
the work yielded by the rope 
(through the space S,) after it 
has passed over the »*, or last 
pulley of the system, U, being 
that done upon it before it 
passes over the first pulley ; 
then by Art 163., equations 




THE PULLEY. 185 

152. 154. and 176., neglecting terms involving powers of 

E D p . 

— , — , - sin. 9 above the first, and observing that a,=a =: 

-p -p -1 / 

&C.=1+-, ct i = a 2 = &C.=~ J l = a a = &c. = -|D-|-(W + fi*) 
<2 (Z # ( 

p sin. (p|, we have 

11,= (l + n-)U n + - D + (W+fi*> sin. 9 S r 



Representing the whole weight of the cord sustained by the 
pulleys by w, and observing that ims—w, we have 

CF - (i + ^)u n + i J nD + (nW + w) ? sin. 9 j S, . . . (177.) 

In the above equations it has been supposed, that although 
the direction of the rope on either side of each pulley is so 
nearly horizontal that cos. 1 may be considered = 0, yet that 
it does so far lend itself over each pulley as to cause the 
surface of the rope to adapt itself to the circumference of 
the pulley, and thereby to produce the whole of that resist- 
ance which is due to the rigidity of the cord. If the tension 
were so great as to cause the cord to rest upon the pulley 
only as a rigid rod or bar would, then must we assume E=0 
and D=0 in the preceding equations. 



174. If one part of the cord passing over a pulley have a 
horizontal, and the other a vertical direction, as, for instance, 
when it passes into the shaft of a mine over the sheaf or 
wheel which overhangs its mouth ; then one of the 

angles i u or » 23 (equation 173.) becomes -, and the 

other or -r, according as the tension on the ver- 
tical cord is downwards or upwards, so that cos. 
i w -hcos. » M =±lj the sign ± being taken according 
as the tension upon the vertical cord is downwards 
or upwards. Moreover, in this case (Art. L70.) 

<=- and cos. 1= — -; therefore (equation 173.) 
4 4/2 




1 1 fl a 1 a{ s 2 ) 



186 



THE PULLEY. 



W P . 



H 1+ ^ + « H u ^l D± w in - ? S s - • • (179) 



174. The modulus of a system of any number of pulleys, over 
one of which the rope passes vertically, and over the rest 
horizontally. 

Let U, repre- 
sent the work 
done upon the 
rope through 
the space S, be- 
fore it passes 
horizontally 
over the first 
pulley of the 
system, and let 
it pass horizon- 
tally over n such pulleys ; and then, after having passed 
over another pulley of different dimensions, let it take a 
vertical direction, descending, for instance, into a shaft. Let 
U 2 be the work yielded by it through the space S, immedi- 
ately that it has assumed this vertical direction : also let ", 
represent the work done upon it in the horizontal direction 
immediately before it passed over this last pulley of the 
system. Then, by equation (179.), 




u, 



a a \ a( V2 ] 



Also, by equation (177.) representing the radius of each of 
the pulleys which carry the rope horizontally by a, the radius 
of its axis by p„ and its weight by W„ and observing that 
U 1 is here the power and u x the work, we have 

U I= (l+— y^-lnV + inW^w^ sin. As,. 

Eliminating the value of u x between these equations, and 

neglecting powers above the first in— , &c, we have 

a 



THE PIVOT. 



187 



fiE j "W> ,(^W 1 + w;)p 1 



a fa 



+ 



», 



I sin. 9 I S, 



(180.) 




175. If the strings be parallel, and their common 
inclination to the vertical be represented by i, so 
that » 1S = i a3 = i ; then, since in this case L=2<z, we 
have (equation 172.), neglecting terms of more than 

one dimension in _ and !!_, 



a 



1 , E 2 P • 
1 + — +— sin. 

a a 



p.+ 



D 






E 

r cos. i 



sm. 9vTJ 3 + 
a \ a 



in which equation i is to be taken greater or less than -, and 

therefore the sign of cos. i is to be taken (as before explained) 
positively or a negatively, according as the tensions on the 
cords act downwards or upwards. If the tensions are verti- 
cal, »=0 or at, according as they act upwards or downwards, 
so that cos. i= ± 1. The above equations agree in this ease, 
as they ought with equations (131.) and (132.). If the par- 
allel tensions are horizontal, then i=-, and the terms involv- 
ing cos. i in the above equations vanish. 




176. The friction of a prvoT. 



When an axis rests upon its bearings, 
not by its convex circumference, bnl by 
its extremity, as shown in the accompany- 
ing figure, it is called a pivot. Lei W 
represent the pressure borne by Buch a 
pivot supposed to act in a direction per- 
pendicular to it- Burface, and to press 



] S8 THE PIVOT. 

equally upon every part of it ; also let p t represent the 
radius of the pivot ; then will *$* represent the area of the 

"W 

pivot, and — - the pressure sustained by each unit of that 

*?? 
area. And if f represent the co-efficient of friction (Art. 

AW 

133.) ? — tL w ill represent the force which must be applied 

parallel to the surface of the pivot to overcome 
q 7 ' ^) the friction of each such unit. Now let the dot- 
ted lines in the accompanying figure represent 
an exceedingly narrow ring of the area of the pivot, and let 
p and p + ^p represent the extreme radii of this ring; then 
will its area be represented by tf(p + Ap) 2 — *p 2 , or by * i2p( A p) + 
(Ap) 2 f , or, since Ap is exceedingly small as compared with p, 
by 2tfpAp. Now the friction upon each unit of this area is 

represented by — z— ; therefore the whole friction upon the 

"VW 2Wf 

ring is represented by — *— . 2tfpAp, or by — ^-pAp, and the 

*Pi Pi 

moment of that friction about the centre of the pivot by 

— Z . p 2 A^ and the sums of the moments of the frictions of 

Pi 2 
all such rings composing the whole area of the pivot by 

2 — f. . p 2 A P , or by — ^2p 9 A P , or by — f- I fd^ or by 

Pi Pi Pi % 

2W/ 

-^HpA or by fWfh (183.); 

whence it appears that the friction of the pivot produces the 
same effect to oppose the revolution of the mass which rests 
upon it, as though the whole pressure which it sustains were 
collected over a point distant by two-thirds of its radius from 
its centre. 

If d represent the angle through which the pivot is made 
to revolve, then -f p/ will represent the space described by 
the point last spoken of ; so that the work expended upon 
the resistance wf acting there, would be represented by 
fWp lt /8, which therefore represents the work expended upon 
the friction of the pivot, whilst it revolves through the angle 



AXES. 189 

6 ; so that the work expended on each complete revolution 
of the pivot is represented by 

frrJW (184)- 



177. If the pivot be hollow, or its surface be an annular 

^^^^^ instead of a continuous circular area, then 

representing its internal radius by p 2 , and 

jlJLJfeJj observing that its area is represented by 

(C^^St^y\ ^(Pi 2- Pa 2 )? an d- therefore the pressure upon 

\w^^ each unit of it by -7—= ^, and the fric- 

\ — =^- «-(p 1 — p 2 ) 

Wf 

tion of each such unit by . a a -, we obtain, as before, 

nPi — p 2 ) 

for the friction of each elementary annulus the expression 

<L. pAp and for the sum of the moments of the frictions 

Pi 2 -P 2 2 

of all the elements of the pivot a _% / J* or 



w@e& 



Let r represent the mean radius of the pivot, i. e. let 
r=-J(p, +p ) ; and let I represent one half the breadth of the 
ring, i.e. let Z=£( Pl — ? a ) ; therefore ?=r+l and p,=r— I. 
Tliese values of p x and p a being substituted in the above for- 
mula, it becomes 

or W/rj 1+4(^)'| (185.); 

whence it follows that the friction of an annular pivot pro- 
duces the same effect as though the whole pressurt u* n 

lectecl over a point in it distant by r j 1 + iM j f rom ;t " 

centre, where r represent its mean radius and I one half its 
breadth. From this it may be shown, as before, that the 



190 -axes. 

whole work expended upon each complete revolution of the 
annular pivot is represented by the formula, 

^fr jl +i(-J }W (186.) 



178. To DETERMINE THE MODULUS OF A SYSTEM OF TWO PRES- 
SURES APPLIED TO A BODY MOVEABLE ABOUT A FIXED AXIS. 
WHEN THE POEST OF APPLICATION OF ONE OF THE-E PRES- 
SURES IS MADE TO REVOLVE WITH THE BODY, THE PERPEN- 
DICULAR DISTANCE OF ITS DIRECTION FROM THE CENTRE RE- 
MAINING CONSTANTLY THE SAME. 

Let the pressures P 1 and P 2 , instead of retaining constantly 
, (as we have hitherto supposed them to do) 
the same relative positions, be now conceived 
continually to alter their relative positions by 
~~ the revolution of the point of application of 
P x with the body, that pressure nevertheless 
retaining constantly the same perpendicular 
distance a from the centre of the axis, whilst 
the direction of P 3 and its amount remain 
constantly the same. 

It is evident that as the point A x thus continually alters its 
position, the distance AjA a or L will continually change, so 
that the value of P x (equation 158.) will continually change. 
Xow the work done under this variable pressure during one 
revolution of P, is represented (Art. 51.) by the formula 

g — fe&dd, if 4 represent the angle A X CA described at 

o 
any time about C, by the perpendicular CA X , and therefore 
aj, the space S described in the same time by the point of 
application A x of P x (see Art. 62.). 

Substituting, therefore, for P x its value from equation 
(158.), we have 




AXES. l\i % 

2tt 



.Mr,=u, + ^y P] . M (mo 

1 

Let now P s be assumed a constant quantity ; 

2- 2tt 

/. - /P 2 Lr$=P 2 a 2 x — /L». 
Now L=A 1 A 2 = {<+2^a 2 cos. 0+a 2 2 }*; 

2- 2t 

o 

o 



neglecting powers of (—4-—) above the first, since in 
\# a <v 

cases its value is less than unity. Integrating this quantity 
between the limits and 2* the second term disappears, so 
that 

2tt 

/L#=(— 9 + — I 2* nearly; 



2tt 

:.P A .i-/L^=PM)(V^=^(Vy ; 

fl,fl,t/ \a, o a / \a, tf a / 



since 2*a 2 is the space through which the point of applica- 
tion of the constant pressure' P a is made to move In each r* 



all 



1.02 AXES. 

volution. Therefore by equation (187), in the case in which 
P a is constant, 

U,=U, j 1+ (p + j,)'r sin- 9 \ (188). 



179. If the pressure P a be supplied by the tension of a 
rope winding upon a drum whose radius is # 3 (as in the cap- 
stan), then is the effect of the rigidity of the rope (Art. 142.) 
the same as though P 3 were increased by it so as to become 



D+EP, 



^^,or (l+-\P 3 + 5 
a 4 \ a J a, 



Now, assuming P, to be constant, and observing that 
U,=2tfP 2 a 2 , we have, by equation (187), 



S* 





Substituting in this equation the above value for P a , 



2tt 



Performing the actual multiplication of these factors, ob- 
serving that — is exceedingly small, and omitting the term 
a, 

involving the product of this quantity and - '■ — , we have 

U ,=P A (l += ) I to+l™l*rLa» } +2*D. 
\ a 9 / ( (i x a^ ft/ ) 

o 

Whence performing the integration as before, we obtain 

U 1 =U,(l + §{l + (l- + l) t ps;n.,} +2 .D. 

If this equation be multiplied by n, and if instead of U, and 
U 9 representing the work done during one complete revolu- 
tion, they be taken to represent the work done through n 
6uch revolutions, then 



193 



[J 1 =U,(l + -) j l + (^+-J*P sin. 9 I +2n*D (189), 



which is the modulus. 



180. K the quantity (- 1 + - ! ) be not so small that terms 

of the binomial expansion involving powers of that quan- 
tity above the first may be neglected, the value of the 

definite integral /L$) may be determined as follows :— 
o 

(0 J 9 + 2a 1 a cos.d + a 2 3 )^=/ {(a l + a t y-2a l a t (l-co*J)\W 



27T a 

=< a »+ a y 1 ^(^Sf ^ ( ' *• Let fc S ' 



2tt 2tt 

:jLM={a t +a,)f[l-& sin. j) * $- 
o o 



=2(a l + a 9 )f(l--V&m. f d)*^*=2(a 1 + a 9 )E 1 (*), where £,(*) 
o 

represents the complete elliptic function of the second order, 
whose modulus is &.* The value of this function is given 
for all values of k in a table which will be found at the end 
of this work. 
Substituting in equation (187), 

U,=U,+^^ . 2(a,+a,) . E,(*)t . P.=U,+ 

CI, 



* See Encyc. Met. art. Def. Int. theorem 2. 

\ An approximate value of Ej(&) is given when k is small by the formula 

E 1 (Ar)=^(]4-K- 1 ), where K = ^r. (See Encyc. Met. art. Def. Int. equation 



on H.) 

13 



lIM- 



TEE CAPST«LN\ 



1/1 1 



U 1= :r,{l + ig4jpBin.,|....(190). 




The Capstan. 

181 The capstan, as used on shipboard, is represented ia 

the accompanying ngure. 
It consists of a solid timber 
CO, pierced through the 
greater part of its length bv 
an aperture AD, which 
receives the upper portion 
of a solid shaft AB of great 
strength, whose lower ex- 
tremity is prolonged, and 
strongly iixed into the rim- 

arms of the capstan, turns upon the snatt 
A.B, resting its weight upon the crown ot 
the =haft. coiling the cable round its cen- 
tral portion CC, and sustaining the teu- 
ton of the cable bv the lateral resistance 
of the shaft. Thus the capstan combines 
the resistances of the pivot and the axis, 
so that the whole resistance to its motion 
h eaual to the sum of the resistances due separately to the 
axk nd the pi vot. and the whole work expended .n turning 
I "eanil to X whole work which would be expended in 
Sit upon its pivot were there no tension ot the cable 
upoh added to the whole work necessary to turn it upon 
I x 'under the tension of the cable were there .no , toction 
„• the pivot. Now, if U, represent the work o be done 
upou the cable in » complete revo utions. the wort .*h ch 
must be done upon the capstan to yield this wo.k upon tbe 
cable is represented (equation 189.) by 




THE CAPSTAN. 195 

where a v represents the length of the arm, and a 2 the radius 
of that portion of the capstan on which the cable is winding. 
Moreover (Art. 176.), the work due to the friction of the 

pivot in n complete revolutions is represented by -nt^fW. 

o 

On the whole, therefore, it appears that the work U, 
expended upon n complete revolutions of the capstan is 
represented by the formula 

u -=( i+ i){ ,+ (?+?r" iB -^^+ 

2^{d+| Pi /wJ.. .(191). 

which is the modulus of the capstan. 

A single pressure P x applied to a single arm has been 
supposed to give motion to the capstan ; in reality, a num- 
ber of such pressures are applied to its different arms when 
it is used to raise the anchor of a ship. These j^essures, 
however, have in all cases,— except in one particular case 
about to be described,— a single resultant. It is that single 
resultant which is to be considered as represented by P„ 
and the distance of its point of application from the axis 
by a l5 when more than one pressure is applied to move the 
capstan. 

The particular case spoken of above, in which the pres- 
sures applied to move the capstan have no resultant, or can- 
not be replaced by any single pressure, is that in which 
they may be divided into two sets of pressure, each set hav- 
ing a resultant, and in which these two resultants are equal, 
act in opposite directions, on opposite sides of the centre, 
perpendicular to the same straight line passing through the 
centre, and at equal distances from it.* 

Suppose that they may be thus compounded into the 
equal pressures R x and E 2 , and let them be replaced 1 »y 
these. The capstan will then be acted upon by four pres- 
sures, — the tension P 2 of the cable, the resistance K of the 
shaft or axis, and the pressures R t and R 5 . Now these pres- 
sures are in equilibrium. If moved, therefore, parallel to 
their present directions, so as to be applied to a single point, 

* Two equal pressures thus placed constitute a statical COUPLB. The pro- 
perties of such couples have been fully discussed by M. Poinsot, and by Mr. 
Pritchard in his Treatise on Statical Couples; some account of them will bo 

found in the Appendix to this work. 



TnK CAPSTAN. 

they would be in equilibrium about that point (Art. 8.). 
Bur when bo removed, It, and R 5 will act in the 

and in opposite direction-. M . they 

are equal to one another ; R and R_ will there:' 
rateJy be in equilibrium with one another when applied to tnat 
point ; and therefore P, and R will se A be in equili- 

brium : whence it follows, that R is equal to P 5 or the whole 

38ure upon the axis, equal in this the wh< ' 

P, upon the cable. So that the friction of the axis is repre- 
sented in every position of the capstan by P. tan. z (tan. o 
being equal to the co-efficient of friction A: - , and 

the work expended on the friction of the ax:- i\ the 

capstan revolves through the angle <? by Pj>£ tan. ^, or by 

P, ( — ) ran. r. or by U t (— I tan. ? : so that, on the whole, 

introducing the correction for rigidity and for the friction of 
the pivot, the modulus (equation 191) becomes in this c 



- rJD+^,/w[....(198> 



This is manifestly the least possible value of the modulus, 
being very nearly that given (equation 191) by the value 
infinity of a^* 

Thus, then, it appears generally from equation (191), that 
the lose friction is less as a 1 is greater, or as P, is applied 
at a greater distance from the axis ; but that it is least of all 
when the pressures are so distributed round the capstan as 
to be reducible to a couple, that case corresponding to the 
value infinity of fl,. This case, in which the moving pres- 
sures upon the capstan are reducible to a .anifestly 
occurs when they are arranged round it in any number of 
pairs, the two pressures of each pair being equal to one an- 
other, acting on opposite sides of the centre, and perpendi- 
cular to the same line passing through it. This sy< 
distribution of rhe pressures about the axis of rhe capstan is 
therefore the most favourable to the working of ir, as well 
as to rhe stability oi rhe shaft which sustains rhe pre- 
upon it. 

* $ being exceedingly small, tan. 6 is very nearly equal to nn. +. 



■Aar 



197 



182. The modulus of a system of three pressures applied 

TO A BODY MOVEABLE ABOUT A CYLINDRICAL AXIS, TWO OF 
THESE PRESSURES BEING GIVEN IN DIRECTION AND PARAL- 
LEL TO ONE ANOTHER, AND THE DIRECTION OF THE THIRD 
CONTINUALLY REVOLTING ABOUT THE AXIS AT THE SAME 
PERPENDICULAR DISTANCE FROM IT. 

Let P 3 and P 3 represent the parallel pressures of the sys- 
p yT ..^ tern, and P x the revolving pressure. 

/: \ From the centre of the axis C, let fall 
/ ; ^ the perpendiculars CA 1? CA 2 , CA 3 upon 
| /$;$ the directions of the pressures, and let 

|LJ'..iA ^ represent the inclination of CAj to 

^ i i CA 3 at any period of the revolution of 

a* i^ P r Let P x be the preponderating 

pressure, and let P 2 act to turn the 
system in the same direction as P l5 and P 3 in the opposite 
direction ; also let R represent the resultant of P 2 and P s , 
and r the perpendicular distance CA of its direction from C. 
Suppose the pressures P 2 and P 3 to be replaced by R ; the 
conditions of the equilibrium of P x throughout its revolu- 
tion, and therefore the work of F 1 will remain unaltered by 
this change, and the system will now be a system of two 
pressures P, and R instead of three ; of which pressures It 
is given in direction. The modulus of this system is there- 
fore represented (equation 187) by the formula 

JJ-JJ r+ L^ilfi .Ldd (193) ; 

o 

where U r represents the work of E, and L represents the dis- 
tance AA t between the feet of the perpendiculars r and a„ 
so that L i =a 1 2 —2a 1 r cosJ + r*=(a 1 —r cos. Vf+'r 1 sin. 2 d ; 

.-. R 2 L 9 =(P^-Rr cos. d) 2 + RV sin. 3 d. 

Now, R=P 3 + P 2 , Kr=P,a,-P a a a ; 

/ .R»L 2 =j(P 3 + P 2 )^-(P A -P^)cos.^ 2 + (P A --P 2 ^ 3 sin.'^ 

[Now if the relations of a x to a a are such that 

| (V t + V t )a l -(P % a t --p t a i ) cos. 6 | l >(P A -P A ytfli. - i 

then the value of ITL 2 will be represented by the Bum of the 



1M AXES. 

squares of two quantities the first of which is greater thar. 
the BeconcL Kd.] Therefore, extracting the square root by 
Poncelet'a theorem, (see Appendix B.) 

RL=a{(P,+P,)« k -.(P A -P A )coB.4} + 3(F t a M -Y t aJ sin.d 
very nearly ; or, 

KL=aa 1 (P 1 + P f )-(P 1 a 1 -P 1 a 1 )(acos. tf — .-3 sin. d). . . .(194). 

e ee 



6 
/*P,«,-P,flr a X a C03 - d ~ 3 sin - *>#> 




(a cos. d — 3 sin. tyft . . . (195). 

If P, and P, be constant, the integral in the second member 
of this equation becomes (P 3 <z 3 — P 2 # 2 ) (a sin. d + 3 cos. •) ; 

whence observing that P 3 # 3 — P 2 # 2 = — 3 "" 3 — 2-2-=— !— — - ; 

also, that TJ r =6Rr=6P t a 9 — t)P 2 a,=U s — U s , and substituting 
in equation (193), we have 

/L£!Z?\( a sin. d + J 3 cos. •)}•••■ ( 196 ) ; 
for complete revolution making •=3«', we have 

reducing, 

r*-t£ +i t)}n....(uD, 

which is the modulus of the system where a and 3 are to be 
lined, as in Note B, (Appendix.) 



THE CHINESE CAPSTAN. 199 

183. If the pressure P 3 be supplied by the tension of a cord 
which winds upon a cylinder or drum at the point A 3 , then 
allowance must be made for the rigidity of the cord, and a 
correction introduced into the preceding equation for that 
purpose. To make this correction let it be observed 
(Art. 142.) that the effect of the rigidity of the cord at A 8 is 
the same as though it increased the tension there from 



R to P, 






or (multiplying both sides of this inequality by #„ and inte- 
grating in respect to &,) as though it increased 

27T 2?r 27T 

Jf'a& to (l +-) yP,a,* + JhdS ; 

o o o 

or, TT 3 to(l+-]IT 3 + MD. 

Thus the effect of the rigidity of the rope to which P 3 is ap- 
plied upon the work U , of that force is to increase it to 

( 1 + — 1 U 3 + 2*D. Substituting this value for U 3 in equa- 
tion (197), and neglecting terms which involve products of 
the exceedingly small quantities—,- ' — ,- — ,andD, 

we have 

l 1 -" i "(^)!' I ' +i! ' D -" <198) ' 

To determine the modulus for n revolutions we must sub- 
stitute in this expression we for «*. 

The Chinese Capstan. 
184. This capstan is represented in the accompanying 




200 THE CHINESE I APSTADT. 

figure under an exceed- 
ingly portable and con- 
venient form.* The axle 
or drum of the capstan 
is composed of two parts 
of different diameters. 
One extremity of the 
cord is coiled upon one 
of these, and the other, in an oposite direction, upon the 
other ; so that when the axle is turned, and the cord is 
wound upon one of these two parts of the drum, it is, at the 
same time, wound off the other, and the intervening cord is 
shortened or lengthened, at each revolution, by as much as 
the circumference of the one cylinder exceeds that of the 
other. In thus passing from one part of the drum to the 
other, the cord is made to pass round a moveable pulley 
which sustains the pressure to be overcome. 

To determine the modulus of this machine, let w 2 and u s 
represent the work done upon the two parts of the cord 
respectively, whilst the work JJ l is done at the moving point 
of the machine, and U 2 yielded at its working point. 

Then, since in this case we have a body moveable about a 
cylindrical axis, and acted upon by three pressures, two of 
which are parallel and constant, viz. the tensions of the two 
parts of the cord ; and the point of application of the third 
is made to revolve about the axis, remaining always at the 
same perpendicular distance from it ; it follows (by equation 
198), that, for n revolutions of the axis, 

U 1 =Aw,— Bw f + 2nflrD (199); 

where 

A=h + - + P sin.p(--- ) Land 



B=-|l- P sin. 9 (- + --^-U 



a % and a a representing the radii of the two parts of the drum, 
a t the constant distance at which the power is applied, and p 
the radius of the axis. 

* A figure of the capstan with a double axle was seen by Dr. 0. Gregory 
among some Chinese drawings more than a century old. It appears to have 
been invented under the particular form shown in the above figure by Mr. G. 
Eckhardt and by Mr. M 4 Lean of Philadelphia. (See Professor Robinson's Mich. 
Phil. vol. ii. p. 255.) 



"HE CHINESE CAPSTAN. 



201 



AlSO, S 



iuce the two 




where 



parts of the cord pass over a pulley, 
and the pulley is made 
to revolve under the ten- 
sions of the two parts of 
the cord, u 3 being the 
work of that tension 
which preponderates, we 
have (by equation 181), 
if S represents the length 
of cord which passes 
over the pulley, 



B 






E 2 Pl . 
— + — -sin. <p 
a a 



, and 



2 W cos. 
a + D 



p x sin. 9 



!■ 



a representing the radius of the pulley, p l the radius of its 
axis, "W its weight, and » the inclination of the direction of 
the tensions of the two parts of the cord to the vertical, the 
axis of the pulley being supposed horizontal, and the two 

_ u z u„ 



parts of the cord parallel. Now t % 



2n*a 9 



— 3-. Sub- 



2 2n*a„ 



stituting these values, and multiplying by 2n*a„ we have 
u 9 a„ 



■=A^ a - 



2weafi x (200). 



Since the tensions t 2 and t s of the two parts of the cord, 
and the pressure P 2 overcome by the machine, are pressures 
applied to the pulley and in equilibrium, and that the points 
of application of t 2 and P 2 are made to move in directions 
opposite to those in which those pressures act, whilst the 
point of application of t 9 is made to move in the same direc- 
tion ; therefore (Art. 59.), 



U 2 + w 2 



■ u. 



\TJ,=u-u,. 



Eliminating w 3 and u z between this equation and equation 



(200), we have 



u. 



AJJ.-ZnvaJS, 



^U Q -27^ 9 B, 



A,-; 



0s 



A.-^ 



202 THE HORSE CAPSTAN. 

Substituting these values in equation (199), and reducing. 



AA-^B 
°t 



(A-B)B 1 g a 



2n* + 2n*D. 



Substituting their values, for A, A„ B, B l5 neglecting terms 
involving more than one dimension of - — — , — , &c. and 
reducing, we obtain for the modulus of the machine, 





^=-\ 



1 + 



E , . (2a 

— hpsm. 9 
a % 






2na,<* 



i », E 2 Pl . 

1— — + — + — sm. <p 



i-u- 



jE— + a(l- r ^)psin.(p} {D-f'Wp 1 cos.»sin.<p} 
a /l_5\ + E + 2 Pl sin.(p 



D 



■2tht.. (201). 



From which expression it is apparent that when the radii &, 
and # 9 of the double axle are nearly equal, a great sacrifice 
of power is made, in the use of this machine, by reason of 
the rigidity of the cord. 



The Horse Capstan, or the Whim Gin. 

185. The whim is a form of the capstan, used in the first 
operations of mining, for raising materials from the shaft and 
levels by the power of horses, before the quantity excavated 
is so great as to require the application of steam power, or 
before the valuable produce of the mine is sufficient to give 
a return upon the expenditure of capital necessary to the 
erection of a steam engine. The construction of this machine 
will be sufficiently understood from the accompanying figure. 
It will bo observed that there are two ropes wound upon the 
drum in opposite directions, and which traverse the space 




THE H0KSE CAPSTAN. 203 

between the capstan 
and the mouth of the 
shaft. One of these 
carries at its extrem- 
ity the descending 
(empty) bucket, and is 
continually in the act 
of winding of! the drum of the capstan as it revolves ; whilst 
the other, from whose extremity is suspended the ascending 
(loaded) bucket, continually winds on the drum. The pres- 
sure exerted by the horses is that necessary to overcome the 
friction of the different bearings, and the other prejudicial 
resistances, and to balance the difference between the weight 
of the ascending load, bucket, and rope, and that of the 
descending bucket and rope. The rope, in passing from the 
capstan to the shaft, traverses (sometimes for a considerable 
distance) a series of sheaves or pulleys, such as those shown 
in the accompanying figure. 

Let now a 2 represent the radius of the drum on which the 
rope is made to wind, and n the number of revolutions 
which it must make to wind up the whole cord ; also let p 
represent the weight of each foot of cord, and & the angle 
which the capstan has described between the time when tiie 
ascending bucket has attained any given position in the 
shaft and that when it left the bottom ; then does a J repre- 
sent the length of the ascending rope wound on the drum, 
and therefore of the descending rope wound off it. Also, 
let "W represent the whole weight of the rope ; then does 
W—is-a^ represent the weight of the ascending rope, and 
paj that of the descending rope, both of which hang sus- 
pended in the shaft. Let P 2 represent the load raised at 
each lift in the bucket, and w the weight of the bucket ; 
then is the tension upon the ascending rope at the mouth of 
the shaft represented by W— pa£ + P 2 + w, and that upon 
the descending rope by paj+w. 

Let, moreover, p z and j? 2 represent the tensions upon these 
ropes after they have passed from the mouth of the shaft, 
over the intervening pulleys, to the circumference of the 
capstan. 

Now, since the tension upon the ascending rope, which is 
W— M.a 9 + P,-f-i0 at the mouth of the shaft, is increased to 
j? 3 at the capstan, and that the tension upon the descending 
rope, which is p 2 at the capstan, is increased to v&£+ vn at 
the mouth of the shaft, if we represent by ( 1 + Ai and B the 
constants which enter into equation L80(Art. 17 1. . we have 



2<'>4: THE HORSE CAPSTAN. 

by that equation (observing that U 1 =P,S 1 and IT^PJS,, so 
that S, disappears from both sides of it), 

^ 3 =(l + A)0V + P 2 + w-^)-f B, .... (202 . 

and h-^ + w=(1 + A)jp 2 + B .... (203). 

Multiplying the former of the above equations by 1-4- A, 
adding them, transposing, dividing by (1-f A), and neglect- 
ing terms of more than one dimension in A and B, 

p-p„=(l +A)(W + P s ) + 2AM> + 2B-2fAa i a. 

Now U r m equation (193) represents the work of the 
resultant of p z and^? 2 during n revolutions of the capstan, it 
therefore equals the difference between the work of p a and 
that of p^ ( see P- 198). 

•*• Vr=Jp 3 a 2 dd -Jp&di = ajlps-p,) dd ; 



2n~ 

;.V r =aJ*{{l+A)(yr + Y,) + 2Aw-T2B-2wft\dd= 
o 

{ (1 + A) (W + P a ) + 2 Aw + 2B | (2n*a,)-^(2n«a q y ; 

/.U r =(l + A)U a +Kl + A)W + 2Aw + 2B-M-S 2 jS a ..(204); 

observing that 2n-« 2 =S„, and that P 2 S 2 =U 2 . 

Now, let it be observed that the pressures applied to the 
capstan are three in number ; two of them, p % and p„ being 
parallel and acting at equal distances a, from its axis ; ana 
the third, P„ hems made to revolve at the constant distance 
a x from the axis (t^ representing the pressure of the horses, 
or the resultant of the pressures of the horses, if there he 
more than one, and a x the distance at which it is applied) ; 
60 that equation 193 (Art. 1S2.) obtains in respect to the 
pressures P„ j>j> 3 , p 2 ; a % being assumed equal to a t . 

Substituting^ and j9 3 for P 2 and P, in equation (194), 

H'L=*a l (j> t i-p 9 )—a t (p % —p i ) (a cos. 6—j3 sin. 6) ; 

2n~ 2;irr 2n~ 

.\ j*Vl.d0=*aJ*(p,+p,) dS - a^p-p,) 



(a cos. 6—,3 sin. 6) dd. 



THE H0ESE CAPSTAN. 



205 



Now, the teiTns of equation (180), represented in the above 
equations by A and B, are all of one dimension in the exceed- 
ingly small quantities D, E, sin. <p. If, therefore, the values 
of Pi and 2> s given by these equations be substituted in the 

2717T 

value of P sm - y I^Ldd (equation 193), then all the terms 



of that expression which involve the quantities A and B will 
be at least of two dimensions in D, E, sin. <p, and may be ne- 
glected. Neglecting, therefore, the values of A and B in 
equations (202, 203), we obtain 

j? 3 +^ 2 =W+P 2 + 2w, and jp,—jp s =W+P 9 — 2^,0 ; 

.". */\jfc+iO de=a, |W+P a + 2w] 2m = {^j { (2maJ P a 

2 

4- ( 2ma t )(W + 2w)}= fe) {S 2 P 2 + S a (W + 2w)\ 
= (j){U 2 + S 2 (W + 2^)h 

representing by S 3 the 
space described by the 
load, and by U, the 
useful work done upon 
it, during n revolu- 
^ tions of the capstan. 

Similarly, 

2nTT 2n7T 

«. Ai>s-i? a )(*cos. 0-/3 sin. 6)dd = a, J { W-t-P,-2f*a f 0} 



(a cos. 0-/3 sin. 0)de=a,(W + F 7 )f{* cos. 0-/3 sin. 0)<#- 



2n.7T 

2f*a 9 " A« cos. 0-/3 sin. 0)0^0. 




-06 ,THE HORSE CAPSTAN. 

2n7r 2n- 



NowAa cog. d—<3 sin. d)dd=(3, and Aa cos. d—fS sin. 6) 



inn 



*« a*f(p-P*)(« cos. 0-/3 sin. a)^=^ f (W + P,)-2i8fto t 
o 

(2n*a 9 )= s «. ^ + ^(^-2^,); observing that P,=^, 

2n7T 

•. f-RLde=a(?i) jU a +S a (W4-2^)i-,^ 2 ^_/3« 2 (W-2fiS 2 ); 



2nr 

psm 



Substituting this value, and also that of U r (equation 204) 
in equation (193), and assuming 

C 1 = (l + A)W + 2Aw + 2B and C,= " (W + 2w)(2i) +2^„ 
we have 

j(-|-Du, + c A -^[ 

e e e 

* For /*0 cos. dd6=d sin. 0— /sin. ddd*=6fi\n. d— vers. 0; also /# sin. 0<i0 




-_ cos. ^ -f- /cos. 6d0* = — 8 cos. 0-|-sin. 0. Now, substituting 2nrr for 0, 


these integrals become respectively and — 2mr. 

• Church's Diff. and Int. Cal. Art. 140. 



THE FRICTION OF CORDS. 20 7 

(C + /3 pfisin.(p \ g _ /3Wj>0 3 sin.<p 
\ a, l 2 a, 

which is the modulus of the machine, all the various ele- 
ments, whence a sacrifice of power may arise in the working 
of it, being taken into account. 

The Friction of Cords. 

186. Let the polygonal line ABC . . . YZ, of an infinite 
_>t number of sides, be taken to represent 
^--* the curved portion of a cord embracing 
Pl any arc of a cylindrical surface (whe- 

t y?\\ \\ ther circular or not), in a plane per- 

:L--'''' N $J ; *• pendicular to the axis of the cylinder ; 

X^^ $.; also let Aa, B&, C<?, &c, be normals 

\*» • or perpendiculars to the curve, inclined 

to one another at equal angles, each 
represented by A0. Imagine the surface of the cylinder to 
be removed between each two of the points A, B, &c, in 
succession, so that the cord may be supported by a small 
portion only of the surface remaining at each of those 
points, whilst in the intermediate space it assumes the direc- 
tion of a straight line joining them, and does not touch the 
surface of the cylinder. Let P x represent the tension upon 
the cord before it has 'passed over the point A ; T, the ten- 
sion upon it after it has passed over that point, or before it 
passes over the point B ; T 2 the tension upon it after it has 
passed over the point B, or before it passes over C ; T s that 
after it has passed over C ; and let P 2 represent the tension 
upon the cord after it has passed over the nth. or last 
point Z. 

Now, any point B of the cord is held at rest by the ten- 
sions T, and T a upon it at that point, in the directions BC 
and BA, and by the resistance R of the surface of the cv Un- 
der there ; and, if we conceive the cord to be there in the 
state bordering upon motion, then (Art. 138.) the direction 
of this resistance K is inclined to the perpendicular MB to 
the surface of the cylinder at an angle KBfl equal to the 
limiting angle of resistance <p. 



208 THE FRICTION OF COEDS. 

Now T„ T„ and E are pressures in equilibrium ; there- 
fore (Art. 1-i.) 

T,_ sin. T,BE 
T 3 ~~sin. T^R ; 

V A0 

but T I BE=AB5-EBJ=J(*-AaB)--EB5 = ^ ~ y -9, 
T,BR=CBJ + EB5=i(«'-BJO)+EBJ)= - - — ^9 ; 

, T ,_ 6iD -{i~ff~ ? )} _ C0S -ff- y ) . 

sin - ji-(y + ? ) S cos -(f- + ? ) 
T _ T cos -(y-*)-«>s.(^ + ?) 

" T. (M \ 

COS. (-g- + 9) 

2 sin. — sm. 9 
2 



A A A 4 

cos cos. 9 — sin. — sin. 9 

2 2 



or dividing numerator and denominator of the fraction in the 

Ad 



second member by cos. -~- cos. 9? 



2 tan. — tan. 9 
T,-T a 2 



1— tan.— tan. 9 

Suppose now the angles Aab, B&C, &c, each of which 
equals Ad, to be exceedingly small, and therefore the points 
A, B, C, &c, to be exceedingly near to one another, and 
exceedingly numerous. By this supposition we shall mani- 
festly approach exceedingly near to the actual case of an in- 
finite number of such points and a continuous surface ; and 



THE FRICTION OF CORDS. 209 

if we suppose Ad infinitely small, our supposition will coincide 
with that case. Now, on the supposition that Ad is exceed- 

Ad 

ingly small, tan. — . tan. <p is exceedingly small, and may 

be neglected as compared with unity ; it may therefore be 
neglected in the denominator of the above fraction. More- 
Ad Ad 
over Ad being exceedingly small, tan. — = — 

T— T 

.-. ' a = tan. <p . Ad* ; /. T 1= T 3 (1 + tan. 9 . Ad). 

Now the number of the points A, B, C, &c. being repre- 
sented by 71, and the whole angle A.dZ between the extreme 
normals at A and Z by d, it follows (Euclid, i. 32.) that 

d 
6r=:n. Ad- therefore Ad=- ; 

T^T^l+^tan.?). 

Similarly, P 1 =T 1 (1 +-tan. 9) 

n 

T,=T,(l+£tan.?), 

&c.=&c.=&c. 

T^_i=P a (l + -tan.?). 

Multiplying these equations together, and striking out fac- 
tors common to both sides of their product, we have 

P^P^l + ^tan. 9)"; 

* If we consider the tension T as a function of d, of which any consecutive 
values are represented by T\ and T 2 , and their difference or the increment of 

T by AT, then ^|^-= tan. f A0; therefore ^ . -^ = - tan. <P; therefore, 
passing to the limit 1 ^ = - tan. <p, and integrating between the limits 
and 0, observing that at the latter limit T=P a , and that at the former it equal* 
P», we have log. (— I = — 6 tan. <f>; therefore P,=P a f '' • 

14 



210 THE FRICTION OF CORDS. 

or P a =P, | 1 + Ti- tan. (?+n— Q a tan. *6- 

or P^P, j 1 + tan. 9 + -— <) 3 tan. "9 + 



KM'-D. 



tan. '9 + 



•!• 



Now this relation of Y l and P 2 obtains however small &6 
be taken, or however great n be taken. Let n be taken 
infinitely great, so that the points A, B, C, &c. may be 
infinitely numerous and infinitely near to each other. The 
supposed case thus passes into the actual case of a con- 

12 3 

tinuous surface, the fractions -,-,-, &c. vanish, and the 
' n 7 n rt 

above equation becomes 

_ _ ( H d tan. 9 d 5 tan. 9 <p s tan. '9 ) 

P 1= P a j 1 + — j— + -T72- +f757f+ ••••)• 

But the quantity within the brackets is the well known ex- 
pansion (by the exponential theorem) of the function £0 tan * ? 

/. P 1 = P 2 e^tan.0 (205). 

Since the length of cord S„ which passes over the point 
A, is the same with that S 2 which passes over the point Z, 
it follows that the modulus (Art. 152.) of such a cylindrical 
surface considered as a machine, and supposed to he fixed 
and to have a rope pulled and made to slip over it$ is 

U I = U 1 e» tan ^ (206). 

It is remarkable that these expressions are wholly inde- 
pendent of the form and dimensions of the surface sustain- 
ing the tension of the rope, and that they depend exclu- 
sively upon the inclination 6 or AdZ of the normals to the 
points A and Z, where the cord leaves the surface, and upon 
the co-efficient of friction (tan. 9), of the material of which 
the rope is composed and the material of which the surface 
is composed. It matters not, for instance, so far as the/Wo 



THE FEICTTCXN" OF COEDS. 211 

tion of the rope or band is concerned, whether it passes 
over a large pulley or drum, or a small one, provided the 
angle subtended by the arc which it embraces is the same, 
and the materials of the pulley and rope the same. 

In the case in which a cord is made to pass m times round 
such a surface, 6=2mn ; 



. p _p £ 2m it tan. 

And this is true whatever be the form of the surface, so 
that the pressure necessary to cause a cord to slip when 
wound completely round such a cylindrical surface a given 
number of times is the same (and is always represented by 
this quantity), whatever may be the form or dimension of 
the surface, provided that its material be the same. It 
matters not whether it be square, or circular, or elliptical. 



187. If P/, P/', P/", &c. represent the pressures which 
must be applied to one extremity of a rope to cause it to 
slip when wound once, twice, three times, &c. round any 
such surface, the same tension P 2 being in each case sup- 
posed to be applied to the other extremity of it, we have 

P/ = P 2 £2;rtan.^ P/'^P^tan.^ p^'^p^tan. ^ & c . = & c . 

So that the pressures P/, P/', P/", &c. are in a geome- 
trical progression, whose common ratio is e 2 ~ tan - tf, which 
ratio is always greater than unity. Thus it appears by the 
experiments of M. Morin (p. 135.), that the co-efficient of 
friction between hempen rope and oak free from unguent is 
•33, when the rope is wetted. In this case tan. <p=-33 and 
2tt tan. ?=2 x 3*14:159 x -33=2-0734:5. The common ratio 
of the progression is therefore in this case e 2 ' 07345 , or it is the 
number whose hyperbolic logarithm is 2*07345. This num- 
ber is 7*95 ; so that each additional coil increases the fric- 
tion nearly eight times. Had the rope been dry, this 
proportion would have been much greater. If an additional 
half coil had been supposed continually to be put upon the 
rope instead of a whole coil, the friction would have been 
found in the same way to increase in geometrical progres- 
sion, but the common ratio would in this case Dave 
been e 7rtan< ? instead of e 2 ^ tan -<A. In the above example the 
value of this ratio would for each half coil have hern 
2-82. 

The enormous increase of friction which results from 



ng. i. jv . a cylind 

»p ya. in jig. 1.. and 

I \ - treniitie- 

. L ,_ ■ _ y upon bv two f 

■"•p *9E: ESi and E'. it ha 

/ ^" to that P i 

4». ^ / rcome R. ■ 



512 THE FRICTION OF CORDS. 

each additional turn of the cord upon a capstan or drum, 
inav from the- ta be understood. 



- We may. from what has been e idily 

explain the reason why a knot connecting the I rtremi- 

nsts the acti 
tend: .._ a em. If a wetted cord be wound round 

n, ; . a cylinder of oak as 

in jig. 1.. and its 

acted 
upon2^y two forces P 
has been 
to that P will not 
/ . Dome R. mV. »i1 

be equal to some- 
where about eight times that force. Xow if the string to 
which R is attached be brought underneath tL -'ring 

so as I seed by it against the surface of the cylinder, 

3. : then, provided the friction produced by 
this pressure be not less than one eighth of P. the string will 
not id ren although the force II cease to act And if 

both extremities of the string be thus made to pass between 
the coil and the cyli: - ssure 

upon each will be requisite, a diminishing the 

radius of the cylinder, this pressure can be increased to any 

since, by a known property of funicular curv 
varies inversely as the radius.* We may. th 
diminish the radius of a c as that : : 

.11 be able to pull away a rope coiled upon it, as 
-ented in Jig. 3., even although one extremity wc-re 
?d upon by no force. 
Fig *- Let us suppoe pe to be 

doubled as in ng. 4.. and coiled 
Then it is apparent, 
from what has been said, that 
the cylinder may be made 
small, that no foreefl P and P 
applied to the extremities of 
f* either of the double cords will 

sufficient to pull them from 
it. in whatever directions these are applied. 




* This property will be proved in that portion of the work which treats of 
the Theory or CoxsTRrcno.v. 



THE FRICTION BREAK. 213 

Now let the cylinder be removed. The cord then being 
drawn tight, instead of being coiled round the cylinder-, wifl 
be coiled round portions of itself, at the points m and n\ 
and instead of being pressed at those points upon the cylin- 
der, by a force acting on one portion of its circumference, it 
will be pressed by a greater force acting all round its cir- 
cumference. All that has been proved before, with regard 
to the impossibility of pulling either of the cords away from 
the coil when the cylinder is inserted, will therefore now 
obtain in a greater degree ; whence it follows that no forces 
P and P' acting to pull the extremities of the cords asunder, 
may be sufficient to separate the knot. 



The Friction Break. 

189. There are certain machines whose motion tends, at 
certain stages, to a destructive acceleration ; as, for instance, 
a crane, which, having raised a heavy weight in one position 
of its beam, allows it to descend by the action of gravity in 
another ; or a railway train, which, on a certain portion of 
its line of transit, descends a gradient, having an inclination 
greater than the limiting angle of resistance. In each of 
these cases, the work done by gravity on the descending 
weight exceeds the work expended on the ordinary resist- 
ance due to the friction of the machine ; and if some other 
resistance were not, under these circumstances, opposed to 
its motion, this excess (of the work done by gravity upon it 
over that expended upon the friction of its rubbing surfaces) 
would be accumulated in it (Art. 130.) under the form of 
vis viva, and be accompanied by a rapid acceleration and a 
destructive velocity of its moving parts. The extraordinary 
resistance required to take up its excess of work, and to 
prevent this accumulation, is sometimes supplied in the 
crane by the work of the laborer, who, to let the weight 
down gradually, exerts upon the revolving crank a pressure 
in a direction' opposite to that which he used in raising it. 
It is more commonly supplied in the crane, and always in 
the railway train, without any work at all of the laborer, by 
a simple pressure of his hand or foot on the lever of the fric- 
tion break, which useful instrument is represented in the 
accompanying figure under the form in which it is com 




21i THE FRICTION BREAK. 

monly applied to the crane, — a form of it which may serve 
to illustrate the principle of its applicai ; '>n under every 

other. BC represents a wheel 
fixed commonly upon that 
axis of the machine to which 
the crank is attached, and 
which axis is carried round 
by it with greater velocity 
than any other. The peri- 
phery of this wheel, which is 
usually of cast iron, is em- 
braced by a strong band* ABCE of wrought iron, fixed 
firmly by its extremity A to the frame of the machine, and 
by its extremity E to the short arm AE of a bent lever PAE, 
which turns upon a fixed axis or fulcrum, at A, and whose 
arm PA, being prolonged, carries a counterpoise D just 
sufficient to overbalance the weight of the arm AP, and to 
relieve the point E of all tension, and loosen the strap from 
the periphery of the wheel, when no force P is applied to the 
extremity of the arm AP, or when the break is out of 
action. 

It is evident that a pressure P applied to the extremity of 
the lever will produce a pressure upon the point E, and a. 
tension upon the band in the direction ABCE, and that 
being fixed at its extremity A, the band will thus be tight- 
ened upon the wheel, producing by its friction a certain 
resistance upon the circumference of the wheel. 

Moreover, it is evident that this resistance of friction upon 
the circumference of the wheel is precisely equal to the 
tension upon the extremity A of the band, being, indeed, 
wholly borne by that tension; and that it is the same 
whether the wheel move, as in this case it does, under the 
band at rest, or whether the band move (under the same 
tensions upon its extremities, but in the opposite direction) 
over the wheel at rest. Let R and Q represent the tensions 
upon the extremities A and E of the band ; then if we sup- 
pose the wheel to be at rest, and the band to be drawn over 
it in the direction ECB by the tension R, and 6 to represent 
the angle subtended at the centre of the wheel by that part 
of its circumference which the band embraces, we have 
(equation 205) 

* Blocks of wood are interposed between the band, the periphery of the 
break wheel. This case will be discussed in the Appendix. 



THE BAND. 215 

Let a x represent the length of the arm AP, and a 9 the 
length of the perpendicular let fall from A npon the direc- 
tion of a tangent to that point in the circumference of the 
wheel where the end EC of the hand leaves it. 

Then, neglecting the friction of the axis A, we have 
(Art. 5.) 

P . a x =Q . 3 ; 



If Pj represent any pressure applied to the circumference of 
the break wheel, and P 2 a pressure applied to the working 
point of the machine, whatever it may be, to which the 
break is applied, and if P x =aP a + 5 (Art. 152.) represent the 
relation between P a and P 2 in the inferior state bordering 
upon motion by the preponderance of P 2 ; then, when P 2 is 
taken in this expression to represent the pressure W, whose 
action upon the working point of the machine the break is 
intended to control, P, will represent that value P of the 
friction upon the break which must be produced by the 
intervention of the lever to control the action of the pressure 
W upon the machine ; so that taking P to represent the 
same quantity as in equation (207), we have 

Eliminating P between this equation and equation (207), 
and solving in respect to P, 



P=^W+J)e-« to * ..... (208). 



The Band. 

190. When the circular motion of any shaft in a machine, 
and the pressure which accompanies that motion, consti- 
tuting together with it the work of the shaft, arc to be com- 
municated to any other distant shaft, this communication L8 



216 



THE BAND. 



usually established by means of a band of 
leather, whieh passes round drums fixed upon 
the two shafts, and has its extremities drawn 
together with a certain pressure and united, 
so as to produce a tension, which should be 
just that necessary to prevent the band from 
slipping upon the drums, subject to the pres- 
sure under which the work is transferred. 
The facility with which this communication 
of rotatory motion may be established or 
broken at any distance and under almost 
every variety of circumstance, has brought 
the band so extensively into use in machinery, 
that it may be considered as a principal chan- 
nel through which work is made to flow in its distribution 
to the successive stages of every process of mechanism, 
carried on in the same workshop or manufactory. 




191. The sum of the tensions upon the two parts of a hand 
is the same, whatever be the pressure under which the band 
is driven, or the resistance overcome, the tension of the 
driving part of the band being always inci^eased by just so 
much as that of the driven part is diminished. 

This principle was first given by M. Poncelet ; it has since 
been amply confirmed by the experiments of M. Morin.* It 
may be proved as followsf : — In the very commencement of 
the motion of that drum to which the driving pressure is 
applied, no motion is communicated by it to the other drum. 
Before any such motion can be communicated to the latter, 
a difference must be produced between the tensions of the 
two parts of the band sufficient to overcome the resistance, 
whatever it may be, which is opposed to the revolution of 
the driven drum. Now, an increase of the tension on the 
driving side of the band must be followed by an elongation 
of that side of the band (since the band is elastic), and by 
the revolution of the circumference of the driving drum 
through a space precisely equal to this elongation. Sup- 
posing, then, the other, or driven side of the band, to 
remain extended, as before, in a straight line between its two 
points of contact with the drums, this portion of the band 



* Nouvelles Experiences sur le Froitement, &c. Metz. 

f No demonstration appears to have been given of it by M. Poncelet. 



THE BAXD. 



217 



must evidently have contracted by precisely the length 
through which the circumference of the drimng drum has 
revolved, or the driving side of the band elongated. Thus, 
the elongation of the driving side of the band is precisely 
equal to the contraction of the driven side. Now, the band 
being supposed perfectly elastic, the increase or dimi- 
nution of its tension is exactly proportional to the increase 
or diminution of its length. The increase of tension on the 
one side, produced by a given elongation, is therefore pre- 
cisely equal to the diminution of tension produced by a con- 
traction equal to that elongation on the other side." Thus, 
if T represent the tension upon each side of the band before 
the driving pressure, whatever it may be, was applied, 
and if T, and T 2 represent the tensions upon the driving 
and the driven sides of the band after that pressure is 
applied; then, since T 1 — T represents the increase of tension 
on the one side, and T— T 2 the diminution of tension on the 
other, T 1 -T=T-T a ; 

.-.T 1 + T 2 =2T (209). 

It is a great principle of the economy of power in the use 
of the band to adjust this initial tension T, so that it may 
just be sufficient to prevent the band from slipping upon 
the drum under any pressure which it is required to transmit. 
The means of making this adjustment will be explained 
hereafter. 

The Modulus of the Baxd. 



Fig\. 



Fig. 2. 



v&y 




192. For simplifying the consideration of this important 
element in machinery, we shall first consider a particular 
case of its application. Let the two drums, whose axis are 
G x and C 2 , be supposed equal to one another, so that the two 
parts of the band which pass round them may be parallel. 

Let, moreover, the centres of the 
two drums be in the same verti- 
cal straight line, so that the two 
parts of the band may be verti- 
cal. 

Let P, and P, be pressures ap- 
plied, in vertical directions, to 
turn the drams, and at perpen- 
dicular distance- from their cen- 
tres, represented by 0,P, and 
C.P,; of which pressures l\ is 
the working or driven pressure, 




218 THE BAND. 

or that which is upon the point of yielding by the prepon 
derance of the other P,. In fig. 1. P 2 is seen applied on 
the same side of the centre of the drums as P„ and in jiff. 
2. on the opposite side. LetTj and T 2 represent the tensions 
upon the two parts of the band, T, being that on the driving, 
and T 2 that on the driven side. 

4=0,^ a f =C,P„ 

r— radius of each drum, 

W = weight of each drum, 

p— radius of axis of each drum, 

Rj and R 2 =resistances of axes of drums, 

<p=limiting angle of resistance. 

Now, the parallel pressures P x , "W, T„ T 2 , R„ applied to the 
lower drum, are in equilibrium ; therefore (Art. 16.), 

E 1 =±(T 1 +T-P-W); 

or substituting for T x + T 2 its value 2T (equation 209), 

^=±(21— Y— W) (210). 

The sign ± being taken according as 2T is greater or less 
than Pj + W, or according as the axis of the lower drum 
presses upon the upper surface of its bearings, as shown in 
fig. I., or upon the lower surface, as shown mfig. 2. In like 
manner, the pressures P 2 , "W, T x , T 2 R 2 , applied to the upper 
drum, being in equilibrium, 

K,=T k +T a qFP a +W, 

or (equation 209) R 2 =2Tq=P 2 + "\y .... (211), 

where the sign =F is to be taken according as P 3 is applied 
on the same side of the axis as P x , or on the opposite side. 

Since, moreover, R x and P 2 act, in the state bordering 
upon motion, at perpendicular distances from the centre of 
the axis, which are each represented by p sin. <p (Art. 153.), 
we have, by the principle of the equality of moments, 

P A +T 9 r=T> + R lP sin. 9 ) m ~ 

P f a,+T a r + K > psin.9=T>| ^ 1J )> 

observing that the resultant of all the pressures applied to 
each drum (excepting only the resistance of its axis) must be 
such as would alone communicate motion to it in the direc- 
tion in which it actually moves, and therefore that the re 
sistance of the axis, which is opposite to this resultant, must 
tend to communicate motion to the drum in a direction oppo- 
site to that in which it actually moves. 



THE BAND. 



219 



Subtracting the above equations, and transposing, 
PA— Pa=(Ri + R 2 ) p sin. 9. 
Substituting the values of E x and R 2 from equations (210) 
and (211), we obtain, in the case in which the negative sign 
of Rj is to be taken, or in which 2T is less than P, + W, the 
axis Cj resting upon the lower surface of its collar as shown 
in fig. 2., 

P A -P A =(P 1 TP 2 + 2W) P sin. 9 ; 

and in the case in which the positive sign of R x is to be 
taken, 2T being greater than P a + W, and the axis G 1 press- 
ing against the upper surface of its collar, as shown in. fig. 1., 

P A -PA=(4T-P i qFP 2 )p sin. 9- 

Transposing and reducing, we obtain for the relation be- 
tween the driving and driven pressures in these two cases 
respectively, 



P _p /^2-Fp sin. o>\ 2Wp sin. 9 



Pa^P, 



[ a 2 -\-p sm, (j)\ 
[a,— psin."9/ 
/ ^ 2 Tpsin. (p v 
l^ + psin. 9/ 



+ 



a x — psm. 9 
4Tp sin. 9 



(213), 
(214), 



■p sin. 9/ ' ^-l-psm.9 
and therefore (by equation 121), for the moduli in the two 



cases. 



U^U, 



TL=TL 




2S t W P sin. 9 
a 1 — p sin. 9 



+ 



4S t T P sin. 
a t + p sin. (6 



(215), 



(216). 



In all which equations the sign =F is to be taken according 
as P 2 is applied on the same side of the line Cfi*, joining the 
axis as P n or on the opposite side. 



193. To determine the initial tenswnT upon the band, 80 that 
it may not slip upon the surface of the drum when sub- 
jected to the given resistance opposed to its motion by the 
work. 



220 



THE BAND. 



Suppose the maximum resistance which may, (luring the 

action of the machine, be opposed to the motion 
of the drum to be represented by a pressure P 
applied at a given distance a from it- centre C 3 . 
Suppose, moreover, that the hand has received 
Buch an initial tension T as .shall jusl cause it to 
be on the point of slipping when the motion of 
the drum is subjected to this maximum resist- 
ance ; and let f, and £ 2 be the tensions upon 
the two parts of the band when it is thus 
just in the act of slipping and of overcoming the resistance 
P. Now, the two parts of the band being parallel, it em 
braces one half of the circumference of each drum ; the rela- 
tion between £, and t^ is therefore expressed (equation 205) 
by the equation 




77 tan. a 

t —t e — 1 
t x = £,£"t*n-0,jwhence we obtain J i J — — - _. But t l + t i = 



e + 1 



2T (equation 209), 



/. t-t,=2T 



(It tan. <p \ 
——I 
-K tan. d I 
e + 1/ 



Also, the relation between the resistance P, opposed to the 
motion of the upper drum, and the tensions t x and £, upon 
the two parts of the band, when this resistance is on the 
point of being overcome, is expressed (equation 212) by the 
equation 

Pa-M 3 r+P 2 p sin. (p=t,r ; 

or substituting the value of E a (equation 211), and transpos- 
ing 

Pa + (2T=FP + W) P s in. <?=(t-t,)r ; 

whence, substituting the value of t x — 1» determined above, 
and transposing, we have 

tan. o 



P(aqFp sin. <p) + Wpsin. <p= 



■HCrO 



r— p sm.c 



THE BAXD. 



221 



T=i 



' P(a=Fp sin. <ft) + TV>sin. <ft 

/ rr tan. o \ 



(217). 




194:. The modulus of the hand under its most general form 

The accompanying figure represents an elastic band pass- 
ing round drums of unequal radii, the 
line joining whose centres C x and C 2 
is inclined at any angle to the vertical, 
and which are acted npon by any 
given pressures P x and P 2 , T 1 being 
supposed to be upon the point of giv- 
ing motion to the system. 

Let Tj and T 2 represent the tensions 
upon the two parts of the band, T x be- 
ing that on the driving side. 
a x , a 2 perpendiculars upon the directions of P x and P 2 re- 
spectively. 

1? 2 the inclinations of the directions of P x and P„ to the 
line OA- 

r„ r 2 the radii of the drums. 
V„ W 2 the weights of the drums. 

» the inclination of the line Cfi^ to the vertical, and 2a, the 
inclination of the two parts of the band to one another. 
p x p 2 the radii of the axes of the drums. 
<f> the limiting angle of resistance between the axis of the 
drum and its collar. 

R 1? R 2 the resistances of the collars in which the axes of 
the drums turn in the state bordering upon motion, or the 
resultants of the pressures upon these axes. The perpendi- 
cular distances at which these resistances act from the cen- 
tres of the axes are (Art. 153.) p, sin. <p and p, sin. (p. Since 
the pressures acting upon the lower drum are T„ T,, P„ W„ 
and R,, and that these pressures are in equilibrium, W, act- 
ing through the centre of the axis, and T, and K, acting to 
turn the drum in one direction about the axis, and P, and T, 
to turn it in the opposite direction ; we have, by the princi- 
ple of the equality of moments (Art. 153.), 

T l a l +T i r l =T l r l +'R l p 1 sin. <p. 
And since T l5 T.,, P a , W s , R, are similarly in equilibrium 



ooo 



THE BAND. 



on the upper drum, W, acting through the centre, and P„ 
R . T acting to turn it in one direction, whilst T, acts to 
turn it in the opposite direction. 

.\P 2 tf 2 + T> 2 + R 2 p 2 sin. c=l>,; 

.•.P A -(T 1 -T> 1 =:K I p I 8in.9 ) 

P 2 a 2 -(T-T : >,v=-E 2 o 2 sin. 9 p 

Let T-T,=2t, and T 1 + T 2 =2T, 

.'• PA-^^R.p, sin. o ) 91 R 

To determine the values of R, and R 2 let the pressures 
applied to each drum be resolved (Art. 11.) in directions 
parallel and perpendicular to the line C 2 C 2 ; those applied to 
the lower drum which, being thus resolved, are parallel to 
0,0,, are 

+T X cos. o. i% +T 2 cos. «,. — P, cos. *„ -W, co^. ,. 

those pressures being taken positively which tend to move 
the axis of the drum from 0, towards C 2 , and those nega- 
tively whose tendency is in the opposite direction. 

In like manner the pressures resolved perpendicular to 
CjC, are 

— T, sin. «>, -f T, sin. a 1? +P X sin. d„ — TT X sin. i, 

those pressures being taken negatively whose tendency 
when thus resolved perpendicular to CjC 2 is to bring that 
line nearer to a vertical direction, and those positively whose 
tendency is in the opposite direction. 

Observing that R, is the resultant of all these pressures, 
we have (Art. 11.) 

Bi"= [(T.+TJcos. *—Y x cos. »,— W, cos. «}' + 

{P, sin. «,— (T x — TJ sin. a-¥, sin. i}\ 

Proceeding similarly in respect to the pressures applied to 
the upper drum, we shall obtain 

B,"= JC^+X) cos. a.-V, cos. a. + TV, cos. «}' + 

jP, sin. ^,-f (T~T,) sin. ",-W, sin. ij»: 

or substituting 2T for T.+T,, and 2t for T x — T, 



»,'= j2Tcos. a.-P, cos. J.-W, COS. l{*+ 
{I\ sin. &-2t sin. «-¥, sin. i} s 

R,'= !2T cos. «,— P, cos. f.+W, cos. i{*+ 
jP 2 sin. K + 2t sin. *,— W, sin. i} s 



(219). 



THE BAXD. 



223 



By eliminating E n E 2 , and t between the four equations 
(218) and (219), a relation is determined between the three 
quantities P 1? P„, T. To simplify this elimination let us sup- 
pose that the preceding hypothesis in respect to the direc- 
tions in which the pressures are to be taken positively and 
negatively is so made, that the expressions enclosed within 
the brackets in the above equations (219) and squared may, 
each of them, represent a positive quantity. Let us, more- 
over, suppose the first of the two quantities squared in each 
equation to be considerably greater than the second, or the 
pressure upon the axis of each drum in the direction of the 
line C x C 2 joining their centres, greatly to exceed the pres- 
sure upon it in a direction perpendicular to that line ; an 
hypothesis which will in every practical case be realised. 
These suppositions being made, we obtain, with a sufficient 
degree of approximation, by Poncelet's Theorem*, 

E i =«{2T cos. e^-P, cos. B— W, cos. i| + 
i8{P x sin. 6 1 —2t sin. a,— W, sin. i}, 

R 2 = a|2T COS. a 1 — P 2 COS. # 2 + \V 2 COS. 1} + 

jS {P„ sin. 2 + 2£ sin. a x — W 2 sin. ij. 

Substituting these values of E x and E 2 in equation (218), 
and reducing, we have 

Y 1 a 1 —2t(r 1 —fip 1 sin. a t sin. <p)= 
P, |2aT cos. «x— PA— ^iTi} sin. <p 
P 2 # 2 — 2t(r 2 — /3p 2 sin. a a sin. <p)= 
— p 2 |2a T cos. * t — PA+W 2 y 2 } sin. 9 J 

where P 1 =( a - cos. X — £ sin. 0,), 
/3,=(a cos. e t —P sin. 2 ), 
y i= :(a cos. » + /3 sin. «), 
y 2 =(a cos. 1— P sin. «). 

Eliminating t between these equations, and neglecting 
terms above the first dimension in ^ sin. <p and p, sin. <p, 

( +PAfa— 0Pi sin. a, sin. <p) ) _ 
( — PaC^i— ^Pi sin - a i sin - ?) ) " 
(• + Pl r 2 (2«T cos. «-PA-W lTl ) ) • f221 

J + Pl r 1 (2«T cos. a _P A + W 2 y 2 ) } sin * *'" l™« 

a, being for the most part exceedingly small, the terms 

* See Appendix. 



(220), 



:24 



THE BAND. 



/3p, sin. a t sin. z and Sp, sin. a x sin. 9 may be neglected; we 
shall then obtain by transposition and reduction 



PA 



+ P 1 <V , 9 (l + r — ' sin. 9) 
-P^l-^sin. 9) 



+ 2aT(p 1 7' 9 + p a r 1 )C0S. a i 



— (W^y^-^y^O 



sm. 



(222). 



If this equation be compared with equation (214), it will 
be found to agree with it, mutatis mutandis, except that 
the co-efficient 2a is in that equation 2. This difference 
manifestly results from the approximate character of the 
theorem of Poncelet. 

Substituting the latter co-efficient for the former, multiply- 

p/3 
ing both sides of the equation by (1 — 'sin. <?), neglecting 



p 
terms of more than two dimensions in — , — , and sin. 0, and 

a x a? 

reducing, 

which is the relation between the moving and working 
pressures in the state bordering upon motion. From this 
relation we obtain for the modulus of the band (equation 
121) 



o ( + 2T(p 1 r 9 + Pa r 1 )cos. *, 



(224). 



If the angle 2 be conceived to increase until it exceed 
5, P, will pass to the opposite side of C,C 2 , and (3 3 will 
become negative; whence it is apparent, that equation 



THE EAXD. 225 

(224) agrees with equation (214) in other respects, and in 
the condition of the ambiguous sign. It is moreover appa- 
rent, from the form assumed by the modulus in this case 
and in that of the preceding article, that the greatest 
economy of power is obtained by applying the moving and 
ths- working pressures on the same side of the line Cfi 9 join- 
ing the axes of the drums. This is in fact but a particular 
case of the general principle established in Art. 168. 



195. The initial tension T of the band may be deter- 
mined precisely as in the former case (equation 217). 

Representing by 6 the angle sub- 
tended by the circumference which 
the band embraces on the second 
or driven drum, by P the maxi- 
mum resistance opposed to its mo- 
tion at the distance a, by $ the 
limiting angle of resistance between 
the band and the surface of the 
drum, and by t 1 and £ 2 the tensions 
upon the two parts of the band, 
when its maximum resistance being opposed, it is upon the 
point of slipping ; observing, moreover, that in this case 

tan. $ 

2ft— *0 or %t is represented (Art. 193.) by 2T e ~ q * ; then 

e + ' 1 
substituting in the second of equations (220) this value for * r 
2£, and P and a for P 5 and « Q , and neglecting the exceed- 
ingly small term which involves the product sin. a x sin. <p, 
we have 

(d tan. $ v 
% «„. * 1 ) *•.= -r. 1 2mT cos - ",-Pft+^.r.i sin * 

Also, since a, represents the inclination of the two parts of 
the band to one another; since, moreover, these touch the 
surfaces of the drums, and that represents tlfb inclination 
of the radii drawn from the centre of the lesser dram to the 
touching points, therefore 6=tt—ol v Substituting this value 
of in the above equation, and solving it in respect to T, we 
have 

15 




226 



THE BAND. 




p 2 /3 2 sin. 0) + W 9 p,7„ sin. <p 



p 2 a COS. ct 1 sill. </> 



}•■ 



• (225), 



196. The modulus of the hand when the two parts of it, 
which intervene oetween the drums ', are made to cross one 
another. 




If the directions of the two parts of the band be made 
to cross, as shown in the accompanying 
figure, the moving pressure T x upon the 
second drum is applied to it rn the side 
opposite to that on which it is applied 
when the bands do not cross ; so that in 
this case, in order that the greatest eco- 
nomy of power may be attained (Art. 
168.), the working pressure or resistance 
P 2 should be applied to it on the side 
opposite to that in which it was applied 
in the other case, and therefore on the side of the line C,C 2 , 
opposite to that on which the moving pressure Pj upon the 
first drum is applied. This disposition of the moving and 
working pressures being supposed, and this case being inves- 
tigated by the same steps as the preceding, we shall arrive 
at precisely the same expressions (equations 223 and 224) 
for the relation of the moving and the working pressures, 
and for the modulus. 

In estimating the value of the initial tension T (equation 
225) it will, however, be found, that the angle d, subtended 
at the centre C 2 of the second drum by the arc KML, which 
is embraced by the band, is no longer in this case repre- 
sented by *— a : but by * + a x . This will be evident if we 
consider that the four angles of the quadrilateral figure 
C 2 KIL being equal to four right angles, and its angles at K 
and L being right angles, the remaining angles KIL and 
KC 2 L are ^qual to two right angles, so that lvC 2 L=*— a, ; 
but the angle subtended by KML equals 2<r— KC 2 L; it 
equals therefore ^H-a . If this value be substituted for #— a, 
in equation (225) it becomes 



THE TEETH OF WHEELS. 



227 



T= 



Via— p 3 /3 a sin. <p) + W 2 p 2 sin. yy 2 



(K7i + ai) tan. <p <v 
\x + ai )t*n.* ) r *~t* " C0S ' "i Sil1 - * 
• + 1/ 



(226.) 



Now the fraction in the denominator of this expression 
being essentially greater in value than that in the denomi- 
nator of the preceding (equation 225), it follows that the 
initial tension T, which must be given to the band in order 
that it may transmit the work from the one drum to the 
other under a given resistance P, is less when the two parts 
of the band cross than when they do not, and, therefore, that 
the modulus (equation 22-1) is less ; so that the band is 
worked with the greatest economy of power {other things 
being the same) when the two parts of it which intervene 
between the drums are made to cross one another. Indeed it 
is evident, that since in this case the arc embraced by the 
band on each drum subtends a greater angle than in the 
other case, a less tension of the band in this case than in the 
other is required (Art. 185.) to prevent it from slipping 
under a given resistance, so that the friction upon the axis 
of the drums which results from the tension of the band is 
less in this case than the other, and therefore the work 
expended on that friction less in the same proportion. 



The Teeth of Wheels. 

197. Let A, B represent two circles in contact at D, and 
moveable about fixed centres at C, and C 2 . It 
is evident that if by reason of the friction of 
these two circles upon one another at D any 
motion of rotation given to A be communicated 
to B, the angles PC,D and QC 2 D described in 
the same time by these two circles, will be such 
as will make the arcs PD and QD which they 
subtend at the circumferences of the circles equal to one 
another. Let the angle PCJD* be represented by <\. and the 
angle QC 2 D by d 2 ; also let the radii &J) and 0,1) of the cir- 
cles be represented by r, and r v Now, arc PD— /y\, arc 
QD— r a s ; and since PD=QD, therefore r^ — r.j', ; 




* Or rather the arc which this angle rabtendfl to radius unity. 



228 



THE TEETH OF WHEELS. 



(227). 



The angles described, in the same time, by two circles 
which revolve in contact arc therefore inversely proportional 

to the radii of the circles, so that their angular velocities 
(Art 74.) bear a constant proportion to one another; and if 
one revolves onif ormly, then the other revolves uniformly ; 

it* the angular revolution of the one varies in any proportion, 
then that of the other varies in like proportion. 

When the resistance opposed to the rotation of the driven 
circle or wheel B i> considerable, it is no longer possible to 
give motion to that circle by the friction on its circum- 
ference of the driving circle. It becomes therefore neces- 
sary in the great majority of cases to cause the rotation of 
the driven wheel by some other means than the friction of 
the circumference of the driving wheel. 

One expedient is the band already described, by means of 
which the weels may be made to drive one another at any 
distances of their centres, and under a far greater resistance 
than they could by their mutual contact. "When, however, 
the pressure is considerable, and the wheels may be brought 
into actual contact, the common and the more certain 
method is to transfer the motion 
from one to the other by means of 
projections on the one wheel called 
teeth, which engage in similar pro- 
jections on the other. 

In the construction of these teeth 
the problem to be solved is, to give 
such shapes to their surfaces of mu- 
tual contact, as that the wheels shall 
be made to tnrn by the intervention 
of their teeth precisely as they would by the friction of 
their circumferences. 




198. That it 




is possible to construct teeth which shall 
answer this condition may thus be shown. 
Let inn and m V be two curves, the one 
described on the plane of the circle A, and 
the other on the plane ot the circle B ; and 
let them be such that as the circle A re- 
volves, carrying round with it the circle B, 
by their mutual contact at D, these two 
curves mm and m'n' may continual?// touch 



THE TEETH OF WHEELS. 



229 



one another, altering of course, as they will do continually, 
their relative positions and their point of contact T. 

It is evident that the two circles would be made to 
revolve by the contact of teeth whose edges were of the 
forms of these two curves mn and m'n' precisely as they 
would by their friction upon the circumferences of one 
another at the point D ; for in the former case a certain 
series of points of contact of the circles (infinitely near to 
one another) at D, brings about another given series of points 
of contact (infinitely near to one another) of the curves mn 
and m'n' at T ; and in the latter case the same series of 
points in the curves mn and m'n' brought into contact neces- 
sarily produces the contact of the same series of points in 
the two circumferences of the two circles at D. 



To construct teeth whose surfaces of contact shall possess 
the properties here assigned to the curves mn and m'n' is 
the problem to be solved. Of the solution of this problem 
the following is the fundamental principle : 




199. In order that two circles A and B may be made to 
revolve by the contact of the surfaces mn and m'n' of their 
teeth, precisely as they would by the friction of their cir- 
cumferences, it is necessary, and it is suf- 
ficient, that a line drawn from the point 
of contact T of the teeth to the point of 
contact D of the circumferences shoxdd, in 
every position of the point T, be perpendi- 
cular to the surfaces in contact there, i. e., 
a normal to both the curves mn and m'n' . 

To prove this principle, we must first establish the follow- 
ing lemma: — If two circles M and N be made to revolve 
about the fixed centres E and F by their mu- 
tual contact at L, and if the planes of these 
circles be conceived to be carried round with 
them in this revolution, and a point P on the 
2>lane of M to trace out a curve PQ on the 
plane of N whilst thus revolving, then is this 
curved line PQ precisely the same as would 
have been described on the plane of N by the same point I\ 
if the latter plane, instead of revolving, had remained al 
rest, and the centre E of the circle M having been released 





230 THE TEETH OF WHEELS. 

from its axis, that circle had been made to roll (carrying its 
plane with it) on the circumference of N. For conceive 
to represent a third plane on which the centres of E and F 
are fixed. It is evident that if, whilst the circles M and N 
are revolving'by their mutual contact, the plane O. to which 
their centres are both fixed, be in any way moved, no change 
will thereby be produced in form of the curve PQ, which the 
point P in the plane of M is describing upon the plane of N, 
such a moti«»n being common to both the planes M and N.* 
Now let the direction in which the circle N ifi revolving be 
that shown by the arrow, and its angular velocity uniform; 
and conceive the plane O to be made to revolve about F with 
an angular velocity (Art. 74 > which is equal to that of X, 
but in an opposite direction, communicating 
this angular velocity to M and X. these re- 
volving meantime in respect to one another, 
and by their mutual contact, precisely as i. 
did before. + 

It is clear that the circle X being earned 
round by its own proper motion in one direc- 
tion, and by the motion common to it and the plane O with 
the •> tdar velocity in the opposite direction, will, in 

reality rest in space ; whilst the centre E of the circle M, 
having no motion proper to itself, will revolve with the 
angular velocity of the plane O, and the various other points 
in that circle with angular velocities, compounded of their 
proper velocities, and those which they receive in common 
with the plane O. these velocities neutralising one another 
at the point L of the circle, by which point it is in contact 
with the circle X. ^o that whilst M revolves round X, the 
point L. by which the former circle at any time touches the 
other, is at rest : this quiescent point of the circle M never- 
theless continually varying its position on the circumferences 
of both circles, and the circle 31 being in fact made to roll 
on the circle X at rest. 

Thus, then, it appears, that by communicating a certain 
eominon angular velocity to both the circles 31 and X about 



* Thus for instance, if the circles M and X continue to revolve, we may 
evidently place the whole machine in a ship under sail, in a moving carriage, 
or upon a revolving wheel, without in the least altering the form of the curve, 
which the p»oint P, revolving with the plane of the circle M, is made to trace 
on the plane of N, because the motion we have communicated is common tc 
both these circles. 

4- M and H may be imagined to be placed upon a horizontal wheel 0, first at 
rest, and then made to revolve backwards in respect to the motion of X. 



THE TEETH OF WHEELS. 231 

the centre F, the former circle is made to roll upon the othei 
at rest ; and, moreover, that this common angular velocity 
does not alter the form of the curve PQ, which a point P in 
the plane of the one circle is made to trace upon the plane 
of the other, or, in other words, that the curve traced under 
these circumstances is the same, whether the circles revolve 
round fixed centres by their mutual contact, or whether the 
centre of one circle be released, and it be made to roll upon 
the circumference of the other at rest. 

This lemma being established, the truth of the proposition 
stated at the head of this article becomes evident ; for if M 
roll on the circumference of N, it is evident that P will, at 
any instant, be describing a circle about their point of con- 
tact L.* 

Since then P is describing, at every instant, a circle about 
L when M rolls upon N, N being fixed, and since the curve 
described by P upon this supposition is precisely the same 
as would have been traced by it if the centres of both cir- 
cles had been fixed, and they had turned by their mutual 
contact, it follows that in this last case (when the circles 
revolve about fixed centres by their mutual contact) the 
point P is at any instant of the revolution describing, during 
that instant, an exceedingly small circular arc about the 
point L ; whence it follows that PL is always a perpendicu- 
lar to the curve PQ at the point P, or a normal to it. 

Now let p be a point exceedingly near to T in the curve 
raV, which curve is fixed upon the plane 
of the circle A. It is evident that, as the 
point p passes through its contact with the 
curve mn at T (see Art. 198.), it will be 
made to describe, on the plane of the circle 
B, an exceedingly small portion of that 
curve mn. But the curve which it is 
(under these circumstances) at any instant 
describing upon the plane of B has been shown to be 
always perpendicular to the line DT ; the curve mn is there- 
fore at the point T perpendicular to the line DT ; whence it 
follows that the curve m'n' is also perpendicular to that line, 
and that DT is a normal to loth those curves at T. This is 
the characteristic property of the curves mn and m'n\ so thai 
they may satisfy the condition of a continual contact wit li 



* For either circle may be imagined to be a polygon of an Infinite Dvmbei 

of sides, on one of the angles of which the rolling circle will, at M»y instant, 
be in the act of turning. 




232 THE TEETH OF WHEELS. 

one another, whilst the circles revolve by the contact of 
their circumferences at D, and therefore conversely, so that 
these curves may, by their mutual contact, give to the cir- 
cles the same motion as they would receive from the contact 
of their circumferences. 



200. To describe, by means of circular arcs, the form of a 
tooth on one wheel which shall work truly with a tooth of 
any given form on another wheel. 

Let the wheels be required to revolve by the action of 
their teeth, as they w r ould by the 
contact of the circles ABE and 
ADF, called their pri?nitive ox pitch 
circles. Let AB represent an arc 
of the pitch circle ABE, included 
between any two similar points A 
and-B of consecutive teeth, and let 
AD represent an arc of the pitch 
circle ADF equal to the arc AB, so 
that the points D and B may come 
simultaneously to A, when the cir- 
cles are made to revolve by their 
mutual contact. AB and AD are 
called the pitches of the teeth of the two wheels. Divide 
each of these pitches into the same number of equal parts 
in the points a, b, &c, a', b', &c. ; the points a and a', b and 
V, &c, will then be brought simultaneously to the point A. 
Let mn represent the form of the face of a tooth on the 
wheel, whose centre is C„ with which tooth a corresponding 
tooth on the other wheel is to work truly ; that is to say, 
the tooth on the other wheel, whose centre is C a , is to be cut, 
so that, driving the surface mn, or being driven by it, the 
wheels shall revolve precisely as they would by the con- 
tact of their pitch circles ABE and ADE at A. From A 
measure the least distance A a to the curve mn, and with 
radius Aa and centre A describe a circular arc a/3 on the 
plane of the circle whose centre is C a . From a measure, in 
like manner, the least distance aa*', to the curve mn, and 
with this distance a*' and the centre a, describe a circular 
arc £7, intersecting the arc a/3 in (3. From the point b 
measure similarly the shortest distance b^" to mn, and with 




THE TEETH OF WHEELS. 233 

the centre V and this distance W describe a circular arc 
7(5, intersecting fiy in 7, and so with the other points of 
division. A curve touching these circular arcs aj8, fiy, y6 i 
<fec, will give the true surface or boundary of the tooth.* 

In order to prove this let it be observed, that the shortest 
distance aa' from a given point a to a given curve mn is a 
normal to the curve at the point of in which it meets it ; and 
therefore, that if a circle be struck from this point a with this 
least distance as a radius, then this circle must touch the 
curve in the point a', and the curve and circle have a com- 
mon normal in that point. 

Now the points a and a will be brought by the revolution 
of the pitch circles simultaneously to the point of contact A, 
and the least distance of the curve mn from the point A will 
then be a*\ so that the arc $7 will then be an arc struck 
from the centre A, with this last distance for its radius. This 
circular arc Qy will therefore touch the curve mn in the point 
a' and the line aa', which will then be a line drawn from 
the point of contact A of the two pitch circles to the point 
of contact a' of the two curves mn and mn\ will also be a 
normal to both curves at that point. The circles will there- 
fore at that instant drive one another (Art. 196.) by the con- 
tact of the surfaces mn and rn'n\ precisely as they would by 
the contact of their circumferences. And as every circular 
arc of the curve m'n similar to fiy becomes in its turn the 
acting surface of the tooth, it will, in like manner, at one 
point work truly with a corresponding point of mn, so that 
the circles will thus drive one another truly at as many 
points of the surfaces of their teeth, as there have been taken 
points of division a, b, &c. and arcs a/3, (3y. &cf 

* This method of describing, geometrically, the forms of teeth is given, without 
demonstration, by M. Poncelet in his Mecanigue Industrielle, 3 me partie, Art. 61 >, 
f The greater the number of these points of division, the more accurate the 
form of the tooth. It appears, however, to be sufficient 
in most cases, to take three points of division, or even 
two, where no great accuracy is required. M. Poncelet 
{Mee. Indust. 3™ partie, Art. 60.) has given the following, 
yet easier, method by which the true form of the tooth 
may be approximated to with sufficient accuracy in most 
cases. Suppose the given tooth X upon the one wheel to 
be placed in the position in which it is first to engage or 
disengage from the reguired tooth on the Other Wheel, 
and let Aa and Xb be equal arcs of the pitch circles OX 
the two wheels whose point of contact 18 A. Dran La 
the shortest distance between A and the face of the tooth 

X; join aa\ bisect that line in //'.and draw mn perpendi- 
cular to an intersecting the circumference Aa in n. It 

from the centre n B circular arc I"' described passing 

through the points and a, it will give the required form 

of the tootli nearly. 




23± 



INVOLUTE TEETH. 



Involute Teeth. 





- which these ore the involut \ 
the pitch of the wht i, the 

clth the radii of the} 

Let OE and OF represent the pitch circles of two wheels, 
AG and BH two circles concentric with 
them and having their radii C\A and I 
in the same proportion with the radii I < > 
and C,0 of the pitch circles. Also lei 
and m'n represent the edges of teeth on the 
two wheels struck by the extremities of flexi- 
ble lines unwinding from the circumferences 
of the circles AG and BH respectively. Let 
these teeth be in contact, in any 
of the wheels, in the point T. and from the 
point T draw TA and TB tangents to the 
generating circles GA and BH in the points 
A and B. Then dr.es AT evidently represent the position of 
the flexible line when its extremity was in the act of gene- 
rating the point T in the curve mn / whence it follows, that 
AT is a normal to the curve mm at the point T*; and in 
like manner that BT is a normal to the curve m'n' at the 
same point T. Now the two curves have a 
common tangent at T : therefore their nor- 
mals TA and TB at that point are in the same 
straight line, being both perpendicular to their 
tangent there. Since then ATB is a straight 
line, and that the vertical angles at the point 
o where AB and C.C, intersect are equal, as 
also the right angles at A and B. it follows 
that the triangles AoC. and B<>C are similar, 
and that C :: I A: CB.' But C A : 

\] :: 0,0 : C .'»:.-. * : I . :: C,< ». 
C,0; therefore the points O and o coincide, 
and the straight line AB, which passes through the point of 




* For if the circle be conceived a polygon of an infinite number of sides, it 
is evident that the line, when in the act of unwinding from it at A, is turning 
upon one ot . - hat polygon, an that its extremity is, 

through an infinitely small .. . .re about that point. 



INVOLUTE TEETH. 



235 



contact T of the two teeth, and is perpendicular to the sur- 
faces of both at that point, passes also through the point of 
contact O of the pitch circles of the wheels. Now this is 
true, whatever be the positions of the wheels, and whatever, 
therefore, be the points of contact of the teeth. Thus then 
the condition established in Art. 199. as that necessary and 
sufficient to the true action of the teeth of wheels, viz. " that 
a line drawn from the point of contact to the pitch circles to 
the point of contact of the teeth should be a normal to their 
surfaces at that point, in all the different positions of the 
teeth," obtains in regard to involute teeth. * 

The point of contact T of the teeth moves along the straight- 
line AB, which is drawn touching the generating circles SH 
and AG of the involutes ; this line is what is called the locus 
of the different points of contact. Moreover, this property 
obtains, whatever may be the number of teeth in contact at 
once, so that all the points of contact of the teeth, if there 
be more than one tooth in contact at once, lie always in this 




line ; which is a characteristic, and a most important pro- 
perty of teeth of the involute form. Thus in the above 

* The author proposes the following illustration of the action of involute 
teeth, which he believes to be new. Conceive AB to represent a band passing 
round the circles AG and BH, the wheels would evidently be driven by this 
band precisely as they would by the contact of their pitch circles, since the 
radii of AG and BH are to one another as the radii of the pitch circles. Con- 
ceive, moreover, that the circles BH and AG carry round with them their 
planes as they revolve, and that a tracer is fixed at any point T of the hand, 
tracing, at the same time, lines mn and m'»',upon both planes, as tbej revolve 
beneath it. It is evident that these curves, being traced by the same point, 
must be in contact in all positions of the circles when driveD by the hand, and 
therefore when driven by their mutual contact. The wheels would therefore 
be driven by the contact'of teeth of the forms mn ami m'ri thus traced bj the 
point T of the band precisely as they would by the contact of their pitch oir. 
cles. Now it is easily seen, that the curves mn and m'n\ thus d< scribed by the 
point T of the band, are involutes of the circles A.G and BH. 



236 



EPICYCLOIDAL AND IIYPCCYCLOIDAL TEETII. 



figure, which represents part of two wheels with involute 
teeth, it will be seen that the points r s of contact of the 
teeth are in the same straight line touching the base* of one 
of the involutes, and passing through the point of contact A 
of the pitch circles, as also the points A and b in that touch- 
ing the base of the other. 



Epicycloidal and Hypocycloidal Teetil 




202. If one circle be made to roll externally on the cir- 
cumference of another, and if, whilst this mo- 
tion is taking place, a point in the circumfe- 
rence of the rolling circle be made to trace 
out a curve upon the plane of the fixed circle, 
the curve so generated is called an epicycloid, 
the rolling circle being called the generating 
circle of the epicycloid, and the circle upon 
which it rolls its base. 

If the generating circle, instead of rolling 
on the outside or convex circumference of its 
base, roll on its inside or concave circumfe- 
rence, the curve generated is called the hypocycloid. 

Let PQ and Pll be respectively an epicycloid and a hypo- 
cycloid, having the same generating circle APH, and 
having for their bases the pitch circles AF and AE of two 
wheels. If teeth be cut upon these wheels, whose edges 
coincide with the curves PQ and PK, they will work truly 
with one another ; for let them be in contact at P, and let 
their common generating circle APH be placed so as to 
touch the pitch circles of both wheels at A, then will its cir- 
cumference evidently pass through the point of contact P 
of the teeth : for if it be made to roll through an exceed- 
ingly small angle upon the point A, rolling there upon the 
circumference of both circles, its generating point will 
traverse exceedingly small portions of both curves ; since 
then a given point in the circumference of the circle APH 
is thus shown to be at one and the same time in the perime- 
ters of both the curves PQ and PR, that point must of 
necessity be the point of contact P of the curves ; since, 



* The circles from which the involutes are described are called their bases. 
This cut and that at page 237. are copied from Mr. Hawkins' edition of Camus 
on the Teeth of Wheels. 



EPICYCLOIDAL AND HYPOCYCLOIDAL TEETH. 



237 



moreover, when the circle APH rolls upon the point A, its 
generating point traverses a small portion of the perimeter 
of each of the curves PQ and PR at P, it follows that the 
line AP is a normal to both curves at that point ; for whilst 
the circle APH is rolling through an exceedingly small 
angle upon A, the point P in it, is describing a circle about 
that point whose radius is AP.* Teeth, therefore, whose 
edges are of the forms PQ and PR satisfy the condition 
that the line AP drawn from the point of contact of the 
pitch circles to any point of contact of the teeth is a normal 
to the surfaces of both at that point, which condition has been 
shown (Art. 199.) to be that necessary and sufficient to the 
correct working of the teeth.f 

Thus then it appears, that if an epicycloid be described 




* The circle APH may be considered a polygon of an infinite number of 
sides, on one of the angles of which polygon it may at any instant be con- 
ceived to be turning. 

f The entire demonstration by which it has been here shown that the 
curves generated by a point in the circumference of a given generating circle 
APH rolling upon the convex circumference of one of the pitch circles, and 
upon the concave circumference of the other are proper to form the edges of 
contact of the teeth, is evidently applicable if any other generating curve be 
substituted for APH. It may be shown precisely in the same manner, that 
the curves PQ and TR generated by the rolling of any such curve (not being 
a circle) upon the pitch circles, possess this property, that the line PA drawn 
from any point of their contact to the point of contact of their pitch circles 
is a normal to both, which property is necessary and sufficient to their correct 
action as teeth. This was first demonstrated as a general principle of the con- 
struction of the teeth of wheels by Mr. Airy, in the Cambridge PhiL Trans, 
vol. ii. He has farther shown, that a tooth of any form whatever being out 
upon a wheel, it is possible to find a curve which, rolling upon the pitch circle 
of that wheel, shall by a certain generating point traverse the edge oi the 
given tooth. The curve thus found being made to roll on the circumference 
of the pitch circle of a second wheel, will therefore trace out the form of ■ 
tooth which will work truly with the first This beautiful property involve* 



23 S EPICYCLOID AL AND HTPOCTCLOIDAL TEETH. 

on the plane of one of the wheels with any gereratmg 
circle, and with the pitch circle of that wheel for its base ; 
and if a hypocyeloid be described on the plane of the other 
wheel with the pitch circle of that wheel for its base; and 
if the faces or acting surfaces of the teeth on the two 
weeelfl be cut so as to coincide with this epicycloid and this 
hypocycloid respectively, then will the wheels be driven 
correctly by the intervention of these teeth. Parts of two 
wheels having epicycloidal teeth are represented in the pre- 
ceding figure. 



203. Lemma. — If the diameter of the generating circle of a 
hypocycloid equal the radius of its base, the hypocycloid 
becomes a straight line having the direction of a radius of 

its base. 

Let D and d represent two positions of the centre of such 
a generating circle, and suppose the 
generating point to have been at A in 
the first position, and join AC ; then 
will the generating point be at P in the 
second position, i. e. at the point where 
CA intersects the- circle in its second 
position; for join Qa and Yd, then 
Z Yda= Z PCrf+ Z CPd=2 AG*. _Also 
2^7=CA; ;.2daxYda=2GAxACa; :.daxFda--=CAx 
ACa ; .'. arc A«=arc Va. Since then the arc aT equals 
the arc aA, the point P is that which in the first position 
coincided with A, i. e. P is the generating point ; and this 
is true for all positions of the generating circle ; the gene- 
rating point is therefore always in the straight line AC. 
The edge, therefore, of a hypocycloidal tooth, the diameter 
of whose generating circle equals half the diameter of the 
pitch circle of its wheel, is a straight line whose direction 
is towards the centre of the wheel.* 



the theoretical solution of the problem -which Poncelet has solved by the 
geometrical construction given to Article 200. If the rolling curve be a 
logarithmic spiral, the involute form of tooth will be generated. 

* The following very ingenious application has been made of this property 
of the hypocycloid to convert a circular into an alternate rectilinear motion. 
AB represents a ring of metal, fixed in position, and having teeth cut upon it* 




TO SET OUT THE TEETH OF WHEELS. 



To SET OUT THE TEETH OF TVhEELS. 

204. All the teeth of the same wheel are constructed of 
the same form and of equal dimensions : it would, indeed, 
evidently be impossible to construct two wheels with dif- 
ferent numbers of teeth, which should work truly with one 
another, if all the teeth on each wheel were not thus alike. 

All the teeth of a wheel are therefore set out by the work- 
man from the same pattern or model, and it is in determining 
the form and dimensions of this single pattern or model of 
one or more teeth in reference to the mechanical effects 
which the wheel is to produce, when all its teeth are cut out 
upon it and it receives its proper place in the mechanical 
combination of which it is to form a part, that consists the 
art of the description of the teeth of wheels. 

The mechanical function usually assigned to toothed wheels 
is the transmission of work under an increased or diminished 
velocity. If CD, DE, &c, represent arcs of the pitch circle 

concave circumference. C is the centre of 
a wheel, having teeth cut in its circum- 
ference to work with those upon the circum- 
ference of the ring, and having the diame- 
ter of its pitch circle equal to half that of 
the pitch circle of the teeth of the ring. 
This being the case, it is evident, that if the 
pitch circle of the wheel C were made to 
roll upon that of the ring, any point in its 
circumference would describe a straight line 
passing through the centre D of the ring ; 
but the circle C would roll upon the ring by 
the mutual action of their teeth as it would 
by the contact of their pitch circles ; if the 
circle C then be made to roll upon the ring 
by the intervention of teeth cut upon both, any point in the circumference of 
C will describe a straight line passing through D. Now, conceive C to be thus 
made to roll round the ring by means of a double or forked link CD, between 
the two branches of which the wheel is received, being perforated at their 
extremities by circular apertures, which serve as bearings to the solid axis of 
the wheel. At its other extremity D, this forked link is rigidly connected 
with an axis passing through the centre of the ring, to which axis is commu- 
nicated the circular motion to be converted by the instrument into an alter- 
nating rectilinear motion. This circular motion will thus be made to cany 
the centre C of the wheel round the point D, and at the same time, cause it to 
roll upon the circumference of the ring. Now, conceive the axis C of the 
wheel, which forms part of the wheel itself, to be prolonged beyond the collar 
in which it turns, and to have rigidly fixed upon its extremity a bar CP. It is 
evident that a point P in this bar, whose distance from the axis C ot the wheel 
equals the radius of its pitch circle, will move precisely as a point in the pitch curie 
of the wheel moves, and therefore that it will describe continually a straight 
line passing through the centre D of the ring. This point P receives, there 
fore, the alternating rectilinear motion which it was required to communicate, 




240 



TO SET OUT THE TEETH OF WHEELS. 




of a wheel intercepted between similar points of consecntive 

teeth (the chords of which arcs are called the pitches of the 

teeth), it is evident that all these arcs mnst be equal, since 

the teeth are all equal and similarly placed ; so that each 

tooth of either wheel, as it passes through its contact with a 

corresponding tooth of the other, carries its pitch line through 

the same space CD, over the point of contact C of the pitch 

lines. Since, therefore, the pitch line of the one wheel is 

carried over a space equal to CD, and that of the other over 

a space equal to cd by the contact of any two of their teeth, 

and since the wheels revolve by the contact of their teeth 

as they would by the contact of their pitch circles at C, it 

follows that the arcs CD and cd are equal. Kow let r x and 

7\ represent the radii of the pitch circles of the two wheels, 

then will 2nr 1 and 2rrr^ represent the circumferences of their 

pitch circles ; and if n x and n^ represent the numbers of 

2ttt 2ttt 

teeth cut on them respectively, then CD= —^ and cd=—^- ? , 

Al , 2^, _ 2rrr, . 
therefore, ? 



?ii 



n. 



r t _ ?i, 



(227); 



Therefore the radii of the pitch circles of the two wheels 
must be to one another as the numbers of teeth to be cut 
upon them respectively. 

Again, let m, represent the number of revolutions made 
by the first wheel, whilst ra 2 revolutions are made by the 
second ; then will 2m\m l represent the space described by 



A TRAIN- OF WHEELS. 241 

the circumference of the pitch circle of the first wheel while 
these revolutions are made, and 27rr 2 ra 2 that described by the 
circumference of the pitch circle of the second ; but the 
wheels revolve as though their pitch circles were in contact, 
therefore the circumferences of these circles revolve through 
equal spaces, therefore 27ir 1 m 1 =27TT 2 m 2 ; 

V -i = :-! (228). 

r % m i K J 

The radii of the pitch circles of the wheels are therefore 
inversely as the numbers of revolutions made in the same 
time by them. 

Equating the second members of equations (227) and (228) 

* = ± (229). 

The numbers of revolutions made by the wheels in the same 
time are therefore to one another inversely as the numbers 
of teeth. 



205. In a train of wheels, to determine how many revolutions 
the last wheel makes whilst the first is maMng any given 
number of revolutions. 

When a wheel, driven by another, carries its axis round 

with it, on which axis a third 

wheel is fixed, engaging with and 

giving motion to & fourth, which, 

UHPlr I 1 i\ 4 m ^ e manner, is fixed upon its 

axis, and carries round with it a 
fifth wheel fixed upon the same 
axis, which fifth wheel engages 
with a sixth upon another axis, 
and so on as shown in the above figure, the combination 
forms a train of wheels. Let n„ n n _, n„ . . . n iP represent die 
numbers of teeth in the successive wheels forming BUch a 
train of^ pairs of wheels; and whilst the first wheel is 
making m revolutions, let the second and third (which revolve 
together, being fixed on the same axis) make m } revolutions; 
the fourth and fifth (which, in like manner, revolve together) 
m n revolutions, the sixth and seventh //>,, and BO on ; and let, 
the last or 2p th wheel thus be made to revolve m r times whilst 

10' 




242 A TRAIN OF WHEELS. 

the fii-it r Bin times. Then, since the first wheel which 

- motion to tl - i which has /<, fa 

and that whilst the former maki b revolutions the latter 

makes m. revolutions, therefore (equation 229 », — i = — ; 

and since, while the third wheel (which revolves with the 
second, rnakt- evolutions, the fourth make- m. revolu- 

tions ; therefore. — = — . Siniilarlv, since while the fifth 

771. 

wheel, which has /< s teeth, makes m^ revolution- revolving 
with the fourth), the sixth, which has 7\ % teeth, make- 

lutions : theretore — = — . in like manner — = — . virc. itc. 

//< s 71 ( 7li i 

p — '~ F ~ . Multiplying these equations together, and 

striking out : 8 c >mmon to the numerator and denomi- 

of the first member of the equation which results from 
their multiplication, we obtain 

7np _ n 1 . n, . n, . . . . ft 2 p-i 
77i " n, . n A ...... 2 p 

The factors in the numerator of this fraction represent the 
numbers of teeth in all the driving wheels of this train, 
and those in the denominator the number- : I _:h in the 
m wheels, or foli s as are more commonly 

call- 

If the numbers of teeth in the former be all equal and 
represented bv n.. and the numbers of teeth in the latter 
also equal and represented by n., then 



77i 



I F m 



Having determined what should be the number of teeth 
in each of the wheels which enter into any mechanical 
combination, with a reference to that particular modification 
of the velocity of the revolving parts of the machine which 
I by that wheel,* it remains next to consider, 
what znust be the dimens: - ach tooth of the wlu 

* The reader is referred for a more complete discussion of this subject (which 

belongs more particularly to descriptive mechank - '.ViIUs's Prin- 

of Mechanism, chap. viL, or to Camus on the Teeth of Wheels, bv Haw- 



THE STRENGTH OF TEETH. 24:2. 

that it may be of sufficient strength to transmit the work 
which is destined to pass through it, under that velocity, or 
to bear the pressure which accompanies the transmission of 
that work at that particular velocity ; and it remains further 
to determine, what must be the dimensions of the wheel 
itself consequent upon these dimensions of each tooth, and 
this given number of its teeth. 



206. To determine the pitch of the teeth of a wheel, knowing 
the work to be transmitted by the wheel. 

Let U represent the number of units of work to be trans- 
mitted by the wheel per minute, m the number of revolutions 
to be made by it per minute, n the number of the teeth to 
be cut in it, T the pitch of each tooth in feet, P the pressure 
upon each tooth in pounds. 

Therefore nT represents the circumference of the pitch 
circle of the wheel, and mnT represents the space in feet 
described by it per minute. Now U represents the work 

transmitted by it through this space per minute, therefore — ~s 

represents the mean pressure under which this work is trans- 
mitted (Art. 50.) ; 

••• p =i » 

The pitch T of the teeth would evidently equal twice the 
breadth of each tooth, if the spaces between the teeth were 
equal in width to the teeth. In order that the teeth of 
wheels which act together may engage with one another and 
extricate themselves, with facility, it is however necessary 
that the pitch should exceed twice the breadth of the tool h 
by a quantity which varies according to the accuracy of the 
construction of the wheel from T Lth to T Vth of the breadth.* 

Since the pitch T of the tooth is dependant upon its 
breadth, and that the breadth of the tooth is dependant, by 
the theory of the strength of materials, upon the pressure V 
which it sustains, it is evident that the quantity P in the 
above equation is a function of T. This functionf may be 
assumed of the form 

* For a full discussion of this subject see Professor Willis's Principles of 
Mechanism, Arts. 107-112. 

f See Appendix, on the dimensions of wheels. 



244 THE STRENGTH OF TEETH. 

T=c 4/P (233) ; 

where c is a constant dependant for its amount upon the 
nature of the material out of which the tooth is formed. 
Eliminating P between this equation and the last, and solving 
in respect to T, 






mn 

The number of units of work transmitted by any machine 
per minute is usually represented in horses' power, one 
horse's power being estimated at 33,000 units, so that the 
number of horses' power transmitted by the machine means 
the number of times 33,000 units of work are transmitted by 
it every minute, or the number of times 33,000 must be 
taken to equal the number of units of work transmitted by 
it every minute. If therefore H represent the number of 
horses' power transmitted by the wheel, then U=33,000H. 
Substituting this value in the preceding equation, and repre- 
senting the constant 33,000c 2 by C 3 , we have 



T=Ci/— (234). 

r mn 



The values of the constant C for teeth of different mate- 
rials are given in the Appendix. 



207. To determine the radius of the pitch circle of a wheel 
which shall contain n teeth of a given pitch. 

^< ^» Let AB represent the pitch T of a tooth, 

\ J\^- ...J.... J and let it be supposed to coincide with its 
chord AMB. Let B represent the radius AC 
of the pitch circle, and n the number of teeth 
to be cut upon the wheel. 

Now there are as many pitches in the cir- 
cumference as teeth, therefore the angle ACB 

subtended by each pitch is represented by- 



n 



Also T= 2 AM = 2 AC sin. £ACB=2B sin. - ; 

n 

.\R=£Tcosec.^ (235). 




TO DESCRIBE EPICTCLOIDAL TEETH. 245 



208. To make the pattern of an epicycloids tooth. 

Having determined, as above, 
the pitch of the teeth, and the 
radius of the pitch circle, strike 
an arc of the pitch circle on a 
thin piece of oak board or me- 
tal plate, and, with a fine saw, 
cut the board through along 
the circumference of this cir- 
cle, so as to divide it into two 
parts : one having a convex and 
rc/ / _ Jj \ the other a corresponding con- 

*c V* cave circular edge. Let EF 
N ' represent one of these portions 
of the board, and GH another. 

Describe an arc of the pitch circle upon a second board or 
plate from which the pattern is to be cut. Let MN repre- 
sent this arc. Fix the piece GH upon this board, so that its 
circular edge may accurately coincide with the circumference 
of the arc 1£N". Take, then, a circular plate D of wood or 
metal, of the dimensions which it is proposed to give to the 
generating circle of the epicycloid ; and let a small point of 
steel P be fixed in it, so that this point may project slightly 
from its inferior surface, and accurately coincide with its cir- 
cumference. Having set off the width AB of the tooth, so 
that twice this width increased by from T Vth to T ] jth of that 
width (according to the accuracy of workmanship to be 
attained) may equal the pitch, cause the circle D to roll upon 
the convex edge GK of the board GH, pressing it, at the 
same time, slightly upon the surface of the board on which 
the arc MN is described, and from which the pattern is to he 
cut, having caused the steel point in its circumference first 
of all to coincide with the point A ; an epicycloidal arc A i ' 
will thus be described by the point P upon the surface MN". 
Describe, similarly, an epicycloidal arc BE through the point 
B, and let them meet in E. 

Let the board GHnow be removed, andletEF be applied 
and fixed, so that its concave edge may accurately coincide 
with the circular arc MN. With the Bame circular plate 1) 
pressed upon the concave edge of EF, and made to roll upon 
it, cause the point in its circumference to describe in tike 
manner, upon the surface of the board from which the pat- 
tern is to be cut, a hypococloidal arc l.II passing through the 



346 



TO DESCRIBE EPICYCLOIDAL TEETH. 



point B, and another AI passing through the point A. HE1 
will then represent the form of a tooth, which will work cor- 
rectly (Art. 202.) with the teeth similarly cut npon any other 
wheel ; provided that the pitch of the teeth bo cut upon the 
other wheel be equal to the pitch of the teeth upon this, and 
provided that the same generating circle D be used to strike 
roes upon the two wheels. 



209. To determine the proper lengths of epicycloidal teeth. 

The general forms of the teeth of wheels being determined 
by the method explained in the preceding article, it remains 
to cut them off of such lengths as may cause them succes- 
sively to take up the work from one another, and transmit it 
under the circumstances most favourable to the economy of 
its transmission, and to the durability of the teeth. 

In respect to the economy of the power in its transmission, 
it is customary, for reasons to be assigned hereafter, to pro- 
vide that no tooth of the one wheel should come into action 
with a tooth of the other until both are in the act of passing 
through the line of centres. This condition may be sati>ticl 
in all cases where the numbers of teeth on neither of the 
wheels is exceedingly small, by properly adjusting the 
lengths of the teeth. Let two of the teeth of the wheels be 
in contact at the point A in the line CD, joining the centres 
of the two wheels ; and let the wheel whose centre is C be 
the driving wheel. Let AH be a portion of the circumfe- 
rence of the generating circle of the teeth, then will the 
points A and B, where this circle intersects the edges of the 




teeth and K of the driving wheel, be points of contact 



TO DESCRIBE EPICYCLODDAL TEETH. 247 

with the edges of the teeth M and L of the driven wheel 
(Art. 202.). Now, since each tooth is to conie into action 
only when it conies into the line of centres, it is clear that 
the tooth L must have been driven by K from the time when 
their contact was in the line of centres, until they have come 
into the position shown in the figure, when the point of con- 
tact of the anterior face of the next tooth O of the driving 
wheel with the flank* of the next tooth M of the driven 
wheel has just passed into the line of centres ; and since the 
tooth O is now to take up the task of impelling the driven 
wheel, and the tooth K to yield it, all that portion of the 
last-mentioned tooth which lies beyond the point B may evi- 
dently be removed ; and if it he thus removed, then the tooth 
K, passing out of contact, will manifestly, at that period of 
the motion, yield all the driving strain to the tooth O, as it 
is required to do. In order to cut the pattern tooth of the 

proper length, so as to satisfy 
the proposed condition, we have 
only then to take A.a (see the 
accompanying figure) equal to 
the pitch "of the tooth, and to 
bring the convex circumference 
of the generating circle, so as 
to touch the convex circumfe- 
rence of the arc MX in that 
point a ; the point of intersec- 
tion e of this circle with the 
\f face AE of the tooth will be 
the last acting point of the tooth ; and if a circle be struck 
from the centre of the pitch circle passing through that 
point, all that portion of the tooth which lies beyond this cir- 
cle may be cut off.f 

The length of the tooth on the wheel intended to act with 
this, may be determined in like manner. 

210. In the preceding article we have supposed the same 

generating circle to be used in striking the entire surfaces 
of the teeth on both wheels. It is not however necessary to 

* That portion of the edge of the tooth which is witlunst the pitch circle is 
called its face, that within it its fank. 

\ The point e thus determined will, in some cases, fall beyond the extremity 
E of the tooth. In such cases it is therefore impossible to cut the tooth <»f 
such a length as to satisfy the required conditions, riz. that it shall dnre only 
after it has passed the line of centres. A lull discussion of these Impoftibto 
cases will be found in Professor Willis's work (Arts. 108 104.), 




248 



TO DESCRIBE EPICYCLOID A.L 11.1111. 



the correct working of the teeth, that the Bame circle should 
thus be used in striking the entin Burfaces of ttwo teeth 

which act tog-ether, but only that the gem rating circle of 
every two portions of the two teeth which come into actual 
contact should be the same. Thus the flank of the driving 
tooth and the face of the driven tooth being in contact at 




P in the accompanying figure,* this face of the one tooth 
and flank of the other must be respectively an epicycloid 
and a hypocycloid struck with the same generating circle. 
Again, the face of a driving tooth and the flank of a driven 
tooth being in contact at Q, these, too, must be struck by 
the same generating circle. But it is evidently unnecessary 
that the generating circle used in the second case should be 
the same as that used in the first. Any generating circle 
will satisfy the conditions in either case (Art. 202.), provided 
it be the same for the epicycloid as for the hypocycloid 
which is to act with it. 

According to a general (almost a universal) custom among 
mechanics, two different generating circles are thus used for 
striking the teeth on two wheels which are to act together, 
the diameter of the generating circle for striking the faces 
of the teeth on the one wheel being equal to the radius of 
the pitch circle of the other wheel. Thus if we call the 
wheels A and B, then the epicycloidal faces of the teeth on 
A, and the corresponding hypocycloidal flanks on B, are 
generated by a circle whose diameter is equal to the radius 
of the pitch circle of B. The hypocycloidal flanks of the 
teeth on B thus become straight lines (Art. 203.), whose 
directions are those of radii of that wheel. In like manner, 



• The upper wheel is here supposed to drive the lower. 



TO TXESCEIBE EPICTCLOIDAL TEETH. 

the epicycloidal faces of the teeth on B, and the correspond- 
ing hypocycloidal flanks of the teeth on A, are struck by a 
circle whose diameter is equal to the radius of the pitch cir 
cle of A ; so that the hypocycloidal flanks of the teeth of A 
become in like manner straight lines, whose directions are 
those of radii of the wheel A. By this expedient of using 
two different generating circles, the flanks of the teeth on 
both wheels become straight lines, and the faces only are 
curved. The teeth shown in the above figure are of this 
form. The motive for giving this particular value to the 
generating circle appears to be no other than that saving of 
trouble which is offered by the substitution of a straight for 
a cwved flank of the tooth. A more careful consideration 
of the subject, however, shows that there is no real economy 
of labour in this. In the first place, it renders necessary 
the use of two different generating circles or templets for 
striking the teeth of any given wheel or pinion, the curved 
portions of the teeth of the wheel being struck with a circle 
whose diameter equals half the diameter of the pinion, and 
the curved portions of the teeth of the pinion with a circle 
whose diameter equals half that of the wheel. Now, one 
generating circle would have done for both, had the work- 
man been contented to make the flanks of his teeth of the 
hypocycloidal forms corresponding to it. But there is yet a 
greater practical inconvenience in this method. A wheel 
and pinion thus constructed will only work with one another ; 
neither will work truly any third wheel or pinion of a differ- 
ent number of teeth, although it have the same pitch. Thus 
the wheels A and B having each a given number of teeth, 
and being made to work with one another, will neither of 
them work truly with C of a different number of teeth of 
the same pitch. For that A may work truly with C, the 
face of its teeth must be struck with a generating circle, 
whose diameter is half that of C: but they are struck with 
a circle whose diameter is half that of B; the condition of 
uniform action is not therefore satisfied. Now let us ap- 
pose that the epicycloidal faces, and the hypocycloidal flanks 
of all the teeth A, B, and C had been struck with the same 
generating circle, and that all three had been of the Bame 
pitch, it is clear that any one of them would then have 
worked truly with any other, and that this would have been 
equally true of any number of teeth oi' the same pitch. 
Thus, then, the machinist may, by the use of the same gen- 
erating circle, for all his pattern wheels n\' the same pilch, BO 
construct them, as that any one wheel of that pitch >hall 



250 TO DESCRIBE EPICYCLOIDAL TEETH. 

work with any other. This offers, under many circumstances 
great advantages, especially in the very great reduction of 
the number of patterns which lie will be required to keep. 
There arc, moreover, many cases in which some arrange- 
ment similar to this is indispensable to the true working of 
the wheels, as when one wheel is required (which is often 
the case) to work with two or three others, of different num- 
bers of teeth, A for instance to turn B and C ; by the ordi- 
nary method of construction this combination would be 
impracticable, so that the wheels should work truly. Any 
generating circle common to a whole set of the same pitch, 
satisfying the above condition, it may be asked whether 
there is any other consideration determining the best dimen- 
sions of this circle. There is such a consideration arising 
out of a limitation of the dimensions of the generating circle 
of the hypocycloidal portion of the tooth to a diameter not 
greater than half that of its base. As long as it remains 
within these limits, the hypocycloidal generated by it is of 
that concave form by which the rlank of the tooth is made 
to spread itself, and the base of the tooth to widen ; when 
it exceeds these limits, the rlank of the tooth takes the con- 
vex form, the base of the tooth is thus contracted, and it* 
strength diminished. Since then, the generating circh 
should not have a diameter greater than half that of any oi 
the wheels of the set for which it is used, it will manifestly 
be the greatest which will satisfy this condition when its 
diameter is equal to half that of the least wheel of the set. 
Xow no pinion should have less than twelve or fourteen 
teeth. Half the diameter of a wheel of the proposed pitch, 
which has twelve or fourteen teeth, is then the true diame- 
ter or the generating circle of the set. The above sugges- 
tions are due to Professor Willis.* 



* Professor Willis has suggested a new and very ingenious method of 
striking the teeth of wheels by means of circular arcs. A detailed description 
of this method has been given by him in the Transactions of the Institution 
of Civil Engineers, vol. ii., accompanied by tables, &c, which render its prac- 
tical application exceedingly simple and easy. 



TO DESCRIBE INVOLUTE TEETH. 



251 



211. To DESCRIBE INVOLUTE TEETH. 



Let AD and AG represent the pitch circles of 
two wheels intended to work together. Draw a 
straight line FE through the point of contact A 
of the pitch circles and inclined to the line of 
centres CAB of these wheels at a certain angle 
FAC, the influence of the dimensions of which 
on the action of the teeth will hereafter he ex- 
plained, but which appears usually to be taken 
not less than 80°. * Describe two circles <?EK 
and yFL from the centres B and C, each touching the 
straight line EF. These circles are to be taken as the bases 
from which the involute faces of the teeth are to be struck. 
It is evident (by the similar triangles ACF and AEB) that 
their radii CF and BE will be to one another as the radii 
CA and BA of the pitch circles, so that the condition neces- 
sary (Art. 201.) to the correct action of the teeth of the 
wheels will be satisfied, provided their faces be involutes to 





these two circles. Let AG and AH in the above figure 
represent arcs of the pitch circles of the wheels on an 
enlarged scale, and ^E, /X, corresponding portions of the 
circles eEK and /FL of the preceding figure. Also let \<f 
represent the pitch of one of the teeth of either wheel. 
Through the points A and a describe involutes ef and mn,"\ 



* See Camus on the Teeth of Wheels, by Hawkins, p. 168. 

f Mr. Hawkins recommends the following as a oonrenienl method of striking 
involute teeth, in his edition of " Camus on the Teeth of Wheels," p. 166. Take 
a thin board, or a plate of metal, and reduce its edge M\ BO IS accurately to 



252 TO DESCRIBE INVOLUTE TEETH. 

Let u be the point where the line EF intersects the involute 
mn\ then it* the teeth on the two wheels are to be nearly of 
the same thickness at their bases, bisect the line Kb in c\ or 
if they are to be of different thicknesses, divide the line Ab 
in a ill the same proportion"'', and strike through the point c 
an involute curve //y, similar to ej\ but inclined in the oppo- 
site direction. If the extremity/*/ of the tooth be then cut 
off so that it may just clear the eircumference of the circle 
fL, the true form of the pattern involute tooth will be 
obtained.' 

There are two remarkable properties of involute teeth, by 
the combination of wdiich they are distinguished from teeth 
of all other forms, and cwteris paribus rendered greatly pre- 
ferable to all others. The iirst of these is, that any two 
wheels having teeth of the involute form, and of the same 
pitch,*)* will work correctly together, since the forms of the 
teeth on any one such wheel are entirely independent of 
those on the wheel which is destined to work with it (Art. 
201.) Any two wheels with involute teeth so made to work 
together will revolve precisely as they would by the actual 
contact of two circles, whose radii may be found by divid- 
ing the line joining their centres in the proportion of the 
radii of the generating circles of the involutes. This pro- 
perty involute teeth possess, however, in common with the 
epicycloidal teeth of different wheels, all of which are struck 
with the same generating circle (Art. 210.) The second no 
less important property of involute teeth — a property which 
distinguishes them from teeth of all other forms — is this, 
that they work equally well, however far the centres of the 



coincide with the circular arc 
- :E eE, and let a piece of thin 
watch-spring OR, having two 
projecting points upon it as 
shown at P, and which is of a 
width equal to the thickness of the plate, be fixed upon its edge by means of 
a screw 0. Let the edge of the plate be then made to coincide with the arc 
eE in such a position that, when the spring is stretched, the point P in it may 
coincide with the point from which the tooth is to be struck; and the spring 
being kept continually stretched, and wound or unwound from the circle, the 
involute arc is thus to be described by the point P upon the face of the board 
from which the pattern is to be cut. 

* This rule is given by Mr. Hawkins (p. 170.); it can only be an approxima- 
tion, but may be sufficiently near to the truth for practical purposes. It is to 
be observed that the teeth may have their bases in any other circles thaD 
those, /'L and eE, from which the involutes are struck. 

f Tlie teeth being also of equal thicknesses at their bases, the method of 
ensuring which condition has been explained above. 




THE TEETH OF A BACK AND PES'ION. 253 

wheels are removed asunder from one another ; so that the 
action of the teeth of two wheels is not impaired when 
their axes are displaced by that wearing of their brasses or 

collars, which soon results from a con- 

/jT^\ tinued and a considerable strain. The 

/ /^ C 7 N \ | existence of this property will readily be 

V Wpk ) admitted, if we conceive AG and BH to 

X^jfe/ represent the generating circles or bases 

y^if^\ °f tne teeth, and these to be placed with 
/ /"r*x \ their centres C 1 and C 2 any distance 
/ »/Cj N | asunder, a band AB (p. 235., note) passing 
I \ *» / I round both, and a point T in this band 
V V__^' J generating a curve ?nn, m' n' on the plane 
X^^^^/i of each of the circles as they are made to 
revolve under it. It has been shown that 
these curves mn and m' n will represent the faces of two 
teeth which will work truly with one another ; moreover, 
that these curves are respectively involutes of the two 
circles AG and BH, and are therefore wholly independent 
in respect to their forms of the distances of the centres of 
the circles from one another, depending only on the dimen- 
sions of the circles. Since then the circles would drive at 
any distance correctly by means of the band ; since, more- 
over, at every such distance they would be driven by the 
curves mn and m'n' precisely as by the band ; and since 
these curves would in every such position be the same 
curves, viz. involutes of the two circles, it follows that the 
same involute curves mn and m'n' would drive the circles 
correctly at whatever distances their centres were placed ; 
and, therefore, that involute teeth would drive these wheels 
correctly at whatever distances the axes of those wheels 
were placed. 



* The Teeth of a Rack and Pinion. 

212. To determine the pitch circle of the pinion. Let IF 
represent the distance through which the rack ie to be 
moved by each tooth of the pinion, and let these teeth be 
N in number; then will the rack be moved through the 
space X . II during one complete revolution of the wheeL 
Now the rack and pinion arc to be driven by the action of 
their teeth, as they would by the contact of the circnm- 



254 



THE TEETH OF A RACK AND PINION. 




ference of the pitch circle of the 
pinion with the plane face of the 
rack, so that the space moved through 
by the rack (hiring one complete 
revolution of the pinion must pre- 
cisely equal the circumference of the 
pitch circle of the pinion. If, there- 
fore we call R the radius of the 
pitch circle of the pinion, then 



2*R=X.H; .\R=^;N 



H. 



213. To describe the teeth of tlie 
pinion, those of the rack being 
straight. The properties which have 
been shown to belong to involute 
teeth (Art. 201.) manifestly obtain, 
however great may be the dimensions of the pitch circle 
of their wheels, or whatever disproportion 
may exist between them. Of two wheels 
OF and OE with involute teeth which 
work together, let then the radius of the 
pitch circle of one OF become infinite, its 
circumference will then become a straight 
line represented by the face of a rack. 
Whilst the radius C 2 of the pitch circle 
OF thus becomes infinite, that C 2 B of the 
circle from which its involute teeth are 
struck (bearing a constant ratio to the first) 
will also become infinite, so that the invo- 
lute m'n' will become a straight line* perpendicular to the 
line AB given in position. The involute teeth on the 
wheel OF will thus become straight teeth (see fig. 1.), hav- 
ing their faces perpendicular to the line AB determined by 
drawing through the point a tangent to the circle AC, 
from which the involute teeth of the pinion are struck. If 
the circle AC from which the involute teeth of the pinion 
are struck coincide with its pitch circle, the line AB becomes 




* For it is evident that the extremity of a line of infinite length unwinding 
itself from the circumference of a circle of infinite diameter will describe, 
through a finite space, a Btraight line perpendicular to the circumference of 
the circle. The idea of giving an oblique position to the straight faces of the 
teeth of a rack appears first to have occurred to Professor Willis. 



THE TEETH OF A RACK AND PINION. 



255 



parallel to the face of the rack, and the edges of the teeth 
of the rack perpendicular to its face {fig. 2.). 

Now, the involute teeth of the one wheel have remained 
unaltered, and the truth of their action with teeth of the 
other wheel has not been influenced by that change in the 
dimensions of the pitch circle of the last, which "has con- 
verted it into a rack, and its curved into straight teeth. 
Thus, then, it follows, that straight teeth upon a rack, work 
truly with involute teeth upon a pinion. Indeed it is evi- 



ct) 



(*.) 




i^U 



UWs 




dent, that if from the point of contact P (fig. 2.) of such an 
involute tooth of the pinion with the straight tooth of a 
rack we draw a straight line PQ parallel to the face ah of 
the rack, that straight line w T ill be perpendicular to the 
surfaces of both the teeth at their point of contact P, and 
that being perpendicular to the face of the involute tooth, 
it will also touch the circle of which this tooth is the invo- 
lute in the point A, at which the face ab of the rack would 
touch that circle if they revolved by mutual contact. Thus, 
then, the condition shown in Art. 191). to be necessary and 
sufficient to the correct action of the teeth, namely, that a 
line drawn from their point of contact, at any time, to the 
point of contact of their pitch circles, is satisfied in respect 
to these teeth. Divide, then, the circumference of the 
pitch circle, determined as above (Art. 212.), into K equal 



THE TEETH OF A RACK. AND PINION. 

parts, and describe (Art. 211.) a pattern involute tooth from 
the circumference of the pitch circle, limiting the length ot 
the face of the tooth to a little more than the length 13P of 
the involute curve generated by unwinding a length AF of 
the flexible line equal to the distance II through which the 
rack is to be moved by each tooth of the pinion. The 
straight teeth of the rack are to be cut of the same length, 
and the circumference of the pitch circle and the face ao of 
the rack placed apart from one another by a little more 
than this length. 

It is an objection to this last application of the involute 
form of tooth for a pinion working with a rack, that the 
point P of the straight tooth of the rack upon which it acts 
is always the same, being determined by its intersection with 
a line AP touching the pitch circle, and parallel to the face 
of the rack. The objection does not apply to the preceding, 
the case {fig. 1.) in which the straight faces of each tooth of 
the rack are inclined to one another. By the continual 
action upon a single point of the tooth oV the rack, it is 
liable to an excessive wearing away of its surface. 



214. To describe the teeth of the pinion, the teeth of the rack 
being curved. 

This may be done by giving to the face of the tooth of 



\J 



L_A 




the rack a cycloidal form, and making the face of the tootb 
of the pinion an epicycloid, as will be apparent if we con- 
, ceive the diameter of the circle whose 

centre is C (see fig. p. 236.) to become 
infinite, the other two circles remain- 
ing unaltered. Any finite portion of 
the circumference of this infinite circle 
will then become a straight line. Let 



AE in the accompanying figure repre- 



THE TEETH OF A WHEEL WITH A LANTERN. 257 

6ent such a portion, and let PQ and PR represent, as 
before, curves generated by a point P in the circle whose 
centre is D, when all three circles revolve by their mutual 
contact at A. Then are PR and PQ the true forms of the 
teeth which would drive the circles as they are driven by 
their mutual contact at A (Art. 202). Moreover, the curve 
PQ is the same (Art. 199.) as would be generated by the 
point P in the circumference of APH ; if that circle rolled 
upon the circumference AQF, it is therefore an epicycloid / 
and the curve PR is the same as would be generated by the 
point P, if the circle API! rolled upon the circumference 
or straight line AE, it is therefore a cycloid. Thus then it 
appears, that after the teeth have passed the line of centres, 
when the face of the tooth of the pinion is driving the flank 
of the tooth of the rack, the former must have an epicy- 
cloidal, and the latter a cycloidal form. In like manner, by 
transferring the circle APH to the opposite side of AE, it 
may be shown, that before the teeth have passed the line of 
centres when the flank of the tooth of the pinion is driving 
the face of the tooth of the wheel, the former must have a 
hypocycloidal, and the latter a cycloidal form, the cycloid 
having its curvature in opposite directions on the flank and 
the face of the tooth. The generating circle will be of the 
most convenient dimensions for the description of the teeth 
when its diameter equals the radius of the pitch circle of 
the pinion. The hypocycloidal flank of the tooth of the 
pinion will then pass into a straight flank. The radius of 
the pitch circle of the pinion is determined as in Art. 212., 
and the method of describing its teeth is explained in 
Art. 208. 



£15. The teeth of a wheel working with a lantern or 

TRUNDLE. 

In some descriptions of mill work the ordinary form of 
the toothed wheel is replaced by a contrivance called a lan- 
tern or trundle, formed by two circular discs, which arc con- 
nected with one another by cylindrical columns called 
staves, engaging, like the teeth of a pinion, with the teeth 
of a wheel which the lantern is intended to drive. This 
combination is shown in the following figure. 

It is evident that the teeth on the wheel which works with 
the lantern have their shape determined by the cylindrical 
IT 



25S THE TEETH OF A WHEEL WITH A LANTEHN. 




shape of the staves. Their forms may readily be found by 
the method explained in Art. 200. 

Having determined upon the dimensions of the staves in 
reference to the strain they are to be subjected to, and upon 
the diameters of the pitch circles of the lantern and wheel, 
and also upon the pitch of the teeth ; strike arcs AB and 
AC of these circles, and set off upon them 
the pitches Aa and Ab from the point of 
contact A of the pitch circles (if the teeth 
are first to come into contact in the line 
'^'"? :: /^Jj of centres, if not, set them off from the 

points behind the line of centres where 
the teeth are first to come into contact). 
Describe a circle ae, having its centre in 
AB, passing through a, and having its 
""iameter equal to that of the stave, and divide each of the 
cches Aa and Ab into the same number of equal parts 
^say three). From the points of division A, a, (3 in the 
pitch Aa, measure the shortest distances to the circle ae, and 
with these shortest distances, respectively, describe from the 
points of division 7, cJ of the pitch Ab, circular arcs inter- 
secting one another; a curve ab touching all these circular 
arcs will give the true face of the tooth (Art, 200.). The 
opposite face of the tooth must be struck from similar cen- 
tres, and the base of the tooth must be cut so far within the 
pitch circle as to admit one half of the stave ae when that 
stave pass<*s the line of centres. 




PRESSURES UPON WHEELS. 



259 






216. The relation between two pressures P, and P s 

APPLIED TO TWO TOOTHED WHEELS IN THE STATE BORDER- 
ING UPON MOTION BY THE PREPONDERANCE OF P r 

Let the influence of the weights of the wheels be in the 
first place neglected. Let B and C represent the centres of 
the pitch circles of the wheels, A their point of contact, P 
the point of contact of the driving and driven teeth at any 
period of the motion, HP the direction of the whole 
resultant pressure upon the teeth at their point of contact, 
which resultant pressure is equal and opposite to the resist- 
ance P of the follower to the driver, BM and CX perpen- 
diculars from the centres of the axes of the wheels upon PP ; 
and BD and CE upon the directions of P 2 and P 3 . 

ED=a l , CE=a„ BM=m 1} GN=m v 

BA=r 1 , CA=r f . 

p i5 p 2 =r radii of axes of wheels. 

<p i5 (p 2 r= limiting angles of resistance between the axes of 
the wheels and their bearings. 

Then, since P x and B applied to the wheel whose centre is 




260 RELATION OF THE DRIVING AND WORKING 

B are in the state bordering upon motion by the preponder 
ance of P„ and since a x and m l are the j)erpendiculars on 
the directions of these pressures respectively, we have (equa- 
tion 158) 

where L, represents the length of the line DM joining the 
feet of the perpendiculars BM and BD. 

Again, since R and P a , applied to the wheel whose centre 
is C, are in the state bordering upon motion by the yiehlimj 
of P 9 (Art. 164.), 

where L Q represents the distance NE between the feet of the 
perpendiculars CE and CN. Eliminating R between tkeso 
equations, we have 



p.= 



&■ 



m 



^-(^) 8in -^ 



(238). 



Now let it be observed, that the line AP, drawn from the 
point of contact A of the pitch circles to the point of contact 
P of the teeth is perpendicular to their surfaces at that point 
P, whatever may be the forms of the teeth, provided that 
they act truly with one another (Art. 199.) ; moreover, that 
when the point of contact P has passed the line of centres, 
as shown in the figure, that point is in the act of moving on 
the driven surface Ppfrom the centre C, or from P towards 
£>, so that the friction of that surface is exerted in the opposite 
direction, or from p towards P ; whence it follows that the 
resultant of this friction, and the perpendicular resistance «P 
of the driven tooth upon the driver, has its direction rP 
within the angle aPp and that it is inclined (Art. 141.) to the 
perpendicular aP at an angle aPr equal to the limiting angle 
of resistance. Now this resistance is evidently equal and 
opposite to the resultant pressure upon the surfaces of the 
teeth in the state bordering upon motion ; whence it follows 
that the angle HP A is equal to the limiting angle of resist- 
ance between the surfaces of contact of the teeth. Let this 
angle be represented by p, and let AP=X. Also let the 



PRESSURES UPON WHEELS. 



261 



inclination PAC of AP to the line of centres BC be repre- 
sented by ft. Through A draw An perpendicular to EP, and 
sAt parallel to it. Then, 

m l =BM=Bt + tM=Bt+An=BA sin. BA£+ AP sin. APR. 
Also BA*=BOE=PAC + APK=d + <p; 

.*. m 1 =r 1 sin. (d-f <p)-f X sin. 9 (239) ; 

<n h =CN=Cs-sK=Cs-An=CA sin. OAs-AP sin. APE. 
But As is parallel to PB, therefore CAs=BOE=0+<p; 

.*. m a =^ 8 sin. (0+tp)— X. sin. 9 (240.). 

Substituting these values of m, and m a in the preceding 
equation, 



(.P.... (241). 



T^sin. (d+9) + Xsin. 9 + (l!i_i\ sin. <p t 

?* a sin. (d + 9)— Xsin.9 — (Ei_?)sin. 9 
\ o 3 / 

217. In the preceding investigation the point of contact P 



Hi)- 




• * 



262 RELATION OF THE DRIVING AND WORKING 

of the teeth of the driving and driven wheels is supposed tc 
have passed the line of centres, or to be behind that line ; 
let us now suppose it not to have passed the line of centres, 
or to be before that line. 

It is evident that in this case the point of contact P is ir 
the act of moving upon the surface pYq of the driven tooth 
towards the centre C, or from P towards q, as in the other 
case it hfrom the centre, or from P towards j?. In this case, 
therefore, the friction of the driven surface is exerted in the 
direction qV ; whence it follows, that in this state bordering 
upon motion the direction of the resistance P of the driven 
upon the driving tooth must lie on the other side of the 
normal APQ, being inclined to it at an angle APN equal to 
the limiting angle of resistance. Thus the inclination of P 
to the normal APQ is in both cases the same, but its position 
in respect to that line is in the one case the reverse of its 
position in the other case.** 

The same construction being made as before, 

m 1 =BM=B^ + ^M=B^+A7i=BA.sin.BA^+AP.sin.APO. 
Also BA£=BOP:=BAP-APO=0-p ;f 
.*. m 1 =zr 1 sin. (d— 9) + ^ sin. 9, 

m,=CN=Cs-sN=Cs-An=CA. sin. CAs-AP. sin. APO. 
But As is parallel to PN", 

/. CAs=BOK=:BAP-APO:=d-<p; 
.-. m i =r i sin. (0— 9)— X sin. 9. 
Substituting these values of m 1 and ra 2 in equation (238), 



■-© 



^sin. (&— <p) + Xsin. 9 + 1^— -J sin. (p l 
sin. (d— 9)— Xsin. 9— l^-^lsin. <p a 



P 2 .(242). 



This expression differs from the preceding (equation 2-il) 
only in the substitution of (d — 9) for (d-f 9) in the first terms 
of the numerator and denominator. 

* Hence it follows, that when the point of contact is in the act of crossing 
the line of centres, the direction of the resultant pressure R is passing from 
one side to the other of the perpendicular APQ ; and therefore that when the 
point of contact is in the line of centres, the resultant pressure is perpendicu- 
lar to that line, and the angle BOR a right angle ; a condition which cannot 
however be assumed to obtain approximately in respect to positions of any 
point of contact exceedingly near to the line of centres. 

f The angle 6 being here taken as before to represent the inclination BAP 
of the line AP, joining the point of contact of the pitch circles with the poiut 
of contact of the teeth, to the hue of centres. 



PRESSURES UPON WHEELS. 



263 



Dividing numerator and denominator of the fraction in 
the second member of that equation by sin. (d+9), and 
tlrrowing out the factors r x and r„ we have 



Vta / 



f / T \ "* 

X sin. 9+ (— -\ sin. 9 

1+ . \°- ' 



7\sm. (d + <p) 



X sin.9 + 



m -» 



^p, 



r 2 sin. (d + 9) 



Now it is evident, that if in this fractional expression d— 9 
be substituted for ^ + 9 the numerator will be increased and 
the denominator diminished, so that the value of ¥ 1 corre- 
sponding to any given value of P 2 will be increased. Whence 
it follows, that the resistance to the motion of the wheels by 
the friction of the common surfaces of contact of their teeth 
and of the bearings of their axes is greater when the contact 
of their teeth takes place before than when it takes place, 
at an equal angular distance, behind the line of centres — a 
principle confirmed by the experience of all practical me- 
chanists. 



218. To DETERMLNE THE RELATION OF THE STATE BORDERING 
UPON MOTION BETWEEN THE PRESSURE Y t APPLIED TO THE 
DRIVING WHEEL AND THE RESISTANCE P 2 OPPOSED TO THE 
MOTION OF THE DRIVEN WHEEL, THE WEIGHTS OF THE 
WHEELS BEING TAKEN INTO THE ACCOUNT. 

ISTow let the influence of the weights W x and W 2 of the 
two wheels be taken into the account. The pressures applied 
to each wheel being now three in number instead of two, the 
relations between r x and R, and P 2 and R are determined 
by equation (163) instead of equation (158). Substituting 
W, and W 2 for P 3 in the two cases, we obtain, instead of 
equations (236) and (237), the following, 

p lj , /L lPl \ . { p , M^ . I 
^ 1 ( /L nPo \ . ) ^ MM n . 



P.= 



hi. 



IWA . )-p M 2 W 2 . 

_|__j 8in . 92 |E- T - a7 -p 2 sin.9 



(243) ; 



in which equations M x and M 2 represent certain functions 



264 



RELATION OF THE DRIVING AND WORKING 




determined (Art. 166.) by the inclinations of the pressures 
Pj and P 2 to the vertical. 

Eliminating B. between the above equations, neglecting 
terms above the first dimensions in sin. <p x and sin. cp 2 , and 
multiplying by a x a 2 , 

?A I ™ 2 — siu. <p 3 [ — Pa J *M — — sin. <p, | = 

M^ . , M 2 W 2 . , QAA . 

-^ — x mj x sin.?^^ — -m^ sin. <p 2 (244). 

Substituting the values of m x and m u from equations (239) 
and (240), and neglecting the products of sin. <p, sin. <p x and 
sin. <p 2 , we obtain 

PA J r 2 sin. (d + <p)— X sin. 9 — s i n . p 2 I — 

P 2 a 2 J ^sin.^ + ^+^sin. 9 + -^ sin - <Px | = 



PRESSTTKES UPON WHEELS. 



265 



< V — ^ 2 Pa sm - 9i+V — r J* sm - ^ r sm - (^+ < p) • • • • (2*0.) 

M 

Now (Art. 166.) — x -=m x cos. i^-fA cos. « 23 , where »„ repre- 

sents the inclination Tv r 1 FP 1 of P, to the vertical, and < 23 the 
inclination RrF of R to the vertical.* 

Let the inclination W X BD of the perpendicular upon P x to 
the vertical be represented by a l3 that angle being so mea- 
sured that the pressure Y 1 may tend to increase it ; let a 2 re- 
present, in like manner, the inclination ECG- of CE to the 
vertical; and let /3 represent the inclination ABr of the 
line of centres to the vertical, 

.-. i 1I =W 1 FP 1 =W 1 BD-BDF=a 1 -~ J 



9? 



« 23 =BrF=BOR-OBr=4+9-/3 ; 

M 

V — 1 —m 1 BULfltj-fa, cos. (0 + 9-/3). 



M 

Similarly — 2 : 



:m 



cos. P 2 GH + # 2 cos. R^W^.f Now 



P a GH=ECG + GEC=a 2 + -; and E^W 2 =^-RrF, and 

RrF was before shown to be equal to (0 + 9-/3 ; 



a. 



= — ra 2 sin. a 2 — # 2 cos. (d+9— /3) 



Substituting the values of m l and ra 2 , from equations (239) 
and (240), 



M 

—=/•, sin. (0 + 9) sin. <*,+>. sin. a, sin. 9 + 
1 

^cos. (0 + 9-/3) 

M 

— ?=— r 9 sm. (0+<p) sm. a 2 +Xsin. a 2 sm. 9— 

a, cos. (d+<P— Z 3 ) 



►... (246). 



* See note, p. 172. 

f It is to be observed that the direction of the arrow in the figure repre- 
sents that of the resistance opposed by the driven wheel to the motion of the 
driving wheel, so that the direction of the pressure of the driving upon the 
driven wheel is opposite to that of the arrow. 



266 RELATION OF THE DRIVING AND WORKING 

Let it be supposed that the distances D^M and EX, repre- 
sented by L, and L,, are of finite dimensions, the directions 
of neither of the pressures P, and P a approaching to coinci- 
dence with the direction of R, — a supposition which has been 
virtually made in deducing equation (163) from equation 
(161 . on the former of which equations, equations (243) de- 
pend. And let it be observed that the terms involving sin. 5 
in the above expressions (equations 246) will be of two di- 
mensions in pg c: 2 and 0, when substituted in equation (245), 
and may therefore be negli </< d. Moreover, that in all cases 
the direction of RP is so nearly perpendicular to the line 
of centres BC, that in those terms of equation (245), which 
are multiplied by sin. z 1 and sin. ;„, the angle $ + <?, or BOP, 

may be asssumed= - ; any error which that supposition in- 
volves, exceedingly small in itself, being rendered exceed- 
ingly less by that multiplication. Equations (246") will then 
become 

M. M, . a 

— =r, sin. aj + flj sin. p, — ==— r, sin. a 2 — a, sin. p. 
a, a, 

Substituting these values in the first factor of the seconc. 
member of equation (245), and representing that factor by 
Njv,, we have 

X/^ 2 — — J \rj x (r x sin. a. + a, sin. ,5) sin. ?,— 
-j^ *iPi(T« sin. " 3 + 9 sin. 5 ) sin. a, ; 




N*=-V^ L ( B ^ n - ».+~sul /3)sin. <p x — 
^Hsin.a.-r^ sin. 3) sin. 9, .... ( 247). 

* If the direction of Px be that of a tangent at the point of contact A of 
the wheels, a case of frequent occurrence, the value of In vanishing, that of N T 
would appear to become infinite in this expression. The difficulty will however 
be removed, if we consider that when a x becomes, as in this case, equal to r u 
and the point M is supposed to coincide with A, In becomes a chord of the pitcb 



PRESSURES UPOX WHEELS. 



267 



Substituting !NVy a for the factor, which it represents in 
equation (245), we have 

PAin sio. (^+9)- x sin. 9 2 -p a sin. 9 2 }-P 2 a 3 h sin. (d+<p) + 



X sin. 9 + — sin. 9^ =]S> 1 f 2 sin. (d +9) . 
a, 

Solving this equation in respect to P 1? 

X sin. © + — ^-sm. 9i 
a, 



(248). 



R = 



¥ 5 



1 + 



7* a sin. (d + 9) 



Xsm. 9 + — -sm. <p. 

^ 

r 2 sm. (0 + 9) J 



F 2 + 






. . . L p 2 . 
X sm. 9 H — — sin. 9 a 



r 9 sin. (0+9) 

Whence, performing actual division by the denominators of 
the fractions in the second member of the equation, and 
omitting terms of two dimensions in sin. <p 1? sin. 9 2 , sin. 9 
(observing that 1ST is already of one dimension in those vari- 
ables), we have 

P-^|l + jx(I + I\sin. ? +^ s i„. ?I+ ^sm.^ 
a x r 2 [ \r 1 rj a l r 1 a^ 



circle, and is therefore represented by 2r x sin. £DBA, or 2rj sin. £ (ai-f-/?) ; so 

in 1 ai s i n o 
that ' r i sin, o^-f-sin. /? _2 sin. ^p^-f-ft ) cos. h( a i—0) _ 

L^ ~~ 2r x sin. * (aT£J3j~ 2ri sin. i (ai-f-/?) 

— cos. ^ (aj — /?). 

If, therefore, we take the angle ai=/?, so as to give to Tx the direction of 
a tangent at A, this expression will assume the value, — cos. 0, or- ; so that 
in this ca9e 

VT Wjp! . Wop, . . o 2 . 

N= — — sin. <t>i - — (sin. a 2 -| ein. fi) sm. 9 . 

Ti Li* r x 



THE MODULUS OF A SYSTEM 



- 



.(l+f)JP,+?^ _ 



In tl ;med that the contact of the teeth 

iiind the line of cent. - 



MODULUS OFi SYSTEM OF TWO TOOTHED WHEELS. 

r<j and n % represent the numbers of teeth in the 
driving and driven wheels respectively, and let it be ob- 
I that these number are one to another as the radii of 
3 of the wheels ; then, multiplying both - 

of equation 249 .-^. we shall obtain 

— =PjflL \ 1— •;>•( — — — )sin. :-— - -in. z.-\ — — sii- 

cosec. (4+ ?){• + N .. 

Now let -- represent an exceedingly small increment of 
the angle C, through which the driven wheel is supposed to 
have revolved, after the point of contact P has passed the 
line of centres ; and let it be observed that the first member 

of the above equation is equal to P,tf — — ~. and that — -^v 

'/*. — - r, 

represents the angle described by the driving wheel 

. whilst the driven wheel describes the angle - - 

whence it follow- ).) that P^l — ±l\ represents the 

work ±U 1 done by the driving pressure P„ whilst this angle 
-cribed by the driven wheel, 

•,^=P A h+{x(I+ .-'— -■:. M-^mLfJ 

I rj ajr x af t 

- - ■ — y?r 

- now A^ be conceived infinitely small, so that the first 
member of the above equation may become the differential 

~cient of U,. in ree - Let -he equation, then, 

be integrated between the limits and -i : P,. L . and L,. 
and therefore X (equation 847 1 being conceived to remain 



OF TWO TOOTHED WHEELS. 269 

constant, whilst the angle 4> is described ; we shall then 
obtain the equation 

U,=Pa fU+ { x (- + -) 8^.9+^8^+ — sin. <p,| 

cosec. (d + 9)U++N.S (250), 

where S is taken to represent the arc r^ described by the 
pitch circle of the driven wheel, and therefore by that of the 
driving wheel also, whilst the former revolves through the 
angle 4>. 



220. The modulus of a system of two toothed wheels, 
the number of teeth on the driven wheel being- con- 



INTO ACCOUNT. 

It is evident that the space traversed by the point of con- 
tact of two teeth on the face of either of them is, in this case, 
small as compared with the radius of its pitch circle, and 
that the direction of the resultant pressure P {see Jig. p. 259.) 
upon the teeth is very nearly perpendicular to the line of 
centres BC, whatever may be the particular forms of the 
teeth ; provided only that they be of such forms as will 
cause them to act truly with one another. In this case, 
therefore, the angle BOP represented by & -f 9 is very nearly 

if 
equal to -, and cosec. (0+<p)=l. 

Since, moreover, PP is very nearly perpendicular to the 
line of centres at A, and that the point of contact P of the 
teeth deviates but little from that line, it is evident that the 
line AP represented by X diners but little from an arc of 
the pitch circle of the driven wheel, and that it differs the 
less as the supposition made at the head of this article more 
nearly obtains. Let us suppose 4- to represent the angle 
subtended by this arc at the centre C of the pitch circle of 
the driven wheel, then will the arc itself be represented by 
ryl'j all d therefore 'k=r^ very nearly. Substituting this 
value of X in equation (250), observing that cosec. (d + <p)=l, 

and that — = — (equation 227), and integrating, 

U 1 =Sl + ^(l + "isin.9H-^sin.9 1 + ^Mn.9j 
\ nj a,7\ <v'i 



270 INVOLUTE TEETH. 

P^+Fr,+ (251). 

But the driven or working pressure P 2 being supposed tc 
remain constant, whilst any two given teeth are in action, 
P,^ represents the work L, yielded by that pressure whilst 
those teeth are in contact : also r t -\> represents the space S, 
described by the circumference of the pitch circle of either 
wheel whilst this angle is described. Now let + be con- 
ceived to represent the angle subtended by the pitch of one 
of the teeth of the driven wheel, these teeth being supposed 

to act only behind the line of centres, then += — , ?i 2 repre- 
ss 
senting the number of teeth on the driven wheel, and J+ 

\ nj n a \ nj \n 1 n a r 

: . U 1= i 1 + * (- + -) sin. 9+ ^ sin. <p x + ^ sin. <pA 
\ \n x nj ajr x a 2 r 3 \ 

U a +£T.S (252), 

which relation between the work done at the moving and 
working points, whilst any two given teeth are in contact, is 
evidently also the relation between the work similarly done, 
whilst any given number of teeth are in contact. It is there- 
fore the modulus of any system of two toothed wheels, the 
numbers of whose teeth are considerable. 



221. The modulus of a system of two wheels with invo- 
lute TEETH OF ANY NUMBERS AND DIMENSIONS. 

The locus of the points of contact of the teeth has been 

shown (Art. 201.) to be in this case 

a straight line DE, which passes 

through the point of contact A of 

the pitch circles, and touches the 

^ - f s^ N circles (EF and DG) from which the 

\\ involutes are struck. Let P repre- 

\ \ sent any position of this point of 

/ ; contact, then is AP measured along 

/ / the given line DE the distance re- 

V/-~— ■;■"''[-■''' presented by X in Art. 216., and the 

angle CAD, which is in this case 

constant, is that represented by 6. Since, moreover, the 

point of contact of the teeth moves precisely as a point P 

upon a flexible cord DE, unwinding from the circle EF and 

winding upon DG, would (see note, p. 235.), it is evident 




INVOLUTE TEETH. 



271 



that the distance AP, being that which such a point would 
traverse whilst the pitch circle AH revolved through a cer- 
tain angle +, measured from the line of centres is precisely 
equal to the length of string which would wind upon DG 
whilst this angle is described by it ; or to the arc of that 
circle which subtends the angle +. If, therefore, we repre- 
sent the angle ACD by »j 3 so that CD=CA cos. ACD=r 2 
cos. ?i, then *.==f a + cos. n. Substituting this value for X in 

<x it 

equation (249), and observing that ^+9 = - — 114-9 = - — 

2 n 

T TV 

(11—9), and that — = — , we have 



a x r i 



l + {+(lH — -Icos. y\ sin. 9 + =^ sin. 9. + '■ 
\ nj a x r x 



p_^ . L iPi,^ « . L *P 2 



Bin. 9 2 



sec. (*)— 9) j-P, 



+ : 



(253); 



from which equation we obtain by the same steps as in 
Art. 219, observing that i\ is constant, 

U 1 =-|l+ \*1 — h— I cos. 1 sin. 9+ -^sin. 9. + -^ sin. <p.} 
( \n, nj a x r x a^ 

sec. (>i— <p)|lJ 9 4-K"S (254), 

which is the modulus of a system of two wheels having any 
given numbers of involute teeth. 



n\ 



222. The estvolute tooth of least resistance. 

It is evident that the value of IT, in equation (254), or of 

the work which must be done 
upon the driving wheel to cause 
a given amount U 2 to be yielded 
by the driven wheel is dependent 
for its amount upon the value of 
^\ the co-efficient of I7 2 in the 
V\ second member of that equation; 
j ■ and that this co-efficient, again, is 
/ / dependent for its value (other 
things being the same") upon the 
value of y) representing the angle 
ACD, or its equal the angle DAI, 




272 THE INVOLUTE TOOTH OF LEAST RESISTANCE. 

which the tangent DE to the circles from which the invo- 
lutes arc struck makes with a perpendicular AI to the line 
of centres. Moreover, that the co-efficient X not involving 
this factor y (equation 247), the variation of the value of 
LI,, so far as this angle is concerned, is wholly involved in 
the corresponding variation of the co-efficient of U 2 and 
becomes a minimum witli it; so that the value of *i which 
gives to the function of v\ represented by this co-efficient, its 
minimum value, is the value of it which satisfies the condi- 
tion of the greatest economy of power, and determines that 
inclination DAI of the tangent DE to the perpendicular to 
the line of centres, and those values, therefore, of the radii 
CD and BE of the circles whence the involutes are struck, 
which correspond to the tooth of least resistance. 

To determine the value of v\ which corresponds to a mini- 
mum value of this co-efficient, let the latter be represented 
by u\ then, for the required value of *], 

du rt ^oVu rt 
— =0, and -n>0. 

Let^(i+-)=A,^isin. <p,+ —sin. <p 3 =B; 
\/i 1 nj ' a x r x aj\ 3 

.\w=l + (A cos. -n sin. 9 + B) sec. (v — <p); 
.*. w=l+B sec. (?]— <p) + A sin. 9 cos. y\ sec. (1— 9); 

.*. -T- =B sec. (11—9) tan. (*i— 9)— A sin. 9 j sin. r\ sec. (1—9)— 

cos. v tan. (11—9) sec. (*i— 9)} ; 

.*. -j7=B sec. 3 (^ —( P) sin. (*i— 9)— 



A sin. 9 sec. 2 (>j— 9) {sin. v\ cos. (1— 9)— - cos. v sin. (*i — 9)} ; 

du 

• 

" dn 



du 

-i-=sec. \ri—cp){B sin. ('/]— 9)— A sin. 3 9j (255). 



In order, therefore, that -v-may vanish for any value of 

*1, one of the factors which compose the second member of 
the above equation must vanish for that value of r\ ; but 
this can never be the case in respect to the first factor, for 
the least value of the square of the secant of an arc is the 
square of the radius. K, therefore, the function u admit of 



THE INVOLUTE TOOTH OF LEAST RESISTANCE. 



273 



a minimum value, the second factor of the above equation 
vanishes when it attains that value ; and the corresponding 
value of n is determined by the equation, 

B sin. 0— 9)— A sin. a 9=0 (256). 

or by sin. (*]— <p)=g-sin. a <p or by *i=<p+sin. ( -g-sin. *<pj ; 
or substituting the values of A and B, 

•7T ( — — sin. <p 



>]=(p + sm. 



L 2 P 5 



JUjp . ^ 2r „ 

— sm. 9 1 + sm. <p 3 



(257). 



Now the function ^ admits of a minimum to which this 
value of -n corresponds, provided that when substituted in 

-j-, this value of -q gives to that second differential co-effi- 
cient of u in respect to i\ a positive value. 
Differentiating equation (255), we have 

oVu 

■ji =2 sec. 2 (?i — 9) tan. (?]— 9){B sin. (*)— 9)- 

A sin. '9} +B sec. \n—(p) cos. (11—9) 

But the proposed value of y (equation 256) has been 
shown to be that which, being substituted in the factor {JB 
sin. (*)— 9)— A sin. "9}, will cause it to vanish, and therefore, 

oJu 



with it, the whole of the first term of the value of 



dr- 



It 



corresponds, therefore, to a minimum, if it gives to the 
second term B sec. 2 (i— 9) cos. (*i— 9) a positive value ; or, 
since sec. *(v—<f>) is essentially positive, and B does not 
involve *i, if it gives to cos. (^—9) a positive value, or if 

ie _1 /A \ « A 

*)— 9 <2 or ^ sin - \ 33 sm - 9( PJ <2' or if g sin - *?<! 5 or & 

A sin. 2 9<B ; or if 



£+=) 



Aft 



sin. f 9<-^sin. 9.4 
^1 



sin. 9, (258). 



Tliis condition being satisfied, the value of *i, determined 

18 



274 THE BEST DIVISION OF THE ANGLE OF CONTACT. 

by equation (257), corresponds to a minimum, and deter* 
mines the ln volute tooth of least resistance.* 



223. To DETERMINE IN WHAT PROPORTION THE ANGLE OF 
CONTACT OF EACH TOOTH SHOULD BE DIVIDED BY THE LINE 
OF CENTRES ; OR THROUGH HOW MUCH OF ITS PITCH EACH 
TOOTH SHOULD DRrVE BEFORE AND BEHIND THE LINE OF 
CENTRE8, THAT THE WORK EXPENDED UPON FRICTION MAY 
BE THE LEAST POSSIBLE. 

Let the proportion in which the angle of contact of each 

tooth is divided by the line of centres be represented by a?, 

2* 
so that x — may represent the angular distance from the line 

of centres of a line drawn from the centre of the driven 
wheel to the point of contact of the teeth when they first 

come into action before the line of centres, and (1—x) — 

V J 71, 

the corresponding angular distance behind the line of centres 

when they pass out of contact ; and let it be observed that, 

on this supposition, if U, represent as before the work 

yielded by the driven wheel during the contact of any two 

teeth, a?U 2 will represent the portion of that work done 

before, and (1— aj)u 3 that done behind, the line of centres. 

Then proceeding in respect to equation (253) by the same 

method as was used in deducing from that equation the 

modulus (Equation 254), but integrating first between the 

2* 
limits and x — , in order to determine the work u x done by 

the driving pressure before the point of contact passes the 

2* 
line of centres, and then between the limits and (1—x) — 

to determine the work u 2 done after the point of contact has 
passed the line of centres ; observing moreover, that in the 
former case — <p is to be substituted in sec. (r\— <p) for 9 (Art. 
217.), we have 

* It may easily be shown by eliminating ij between equations (254) and 
(256) that the modulus corresponding to this condition of the greatest economy 
of power, where involute teeth are used, is represented by the formula 

U 1= | 1+*A sin. 20-KB 2 - A 5 sin. *</>) Lrj 2 +NS. 



THE BEST DIVISION OF THE ANGLE OF CONTACT. 275 

u t = J 1+ {**(- + -) cos. n sin- * + ^ Bm - *« + 

-£sin. 9 a |sec. fr+9) f artJ.+N*,; 
Or assuming 

— H — I cos. ii sin. 9=0, and sin. <p,H sin. 9 a =0 

^ 1 =|l + («aj-|-5)sec. (*i+9)|»XJ a +N* I . 

*, representing the space described by the pitch circle of 
either wheel before the line of centres is passed ; similarly, 

u,= | 1+ \ a (l-x) + b} sec. (ij— 9) \ (l-aOU.+N*.. 

Adding these equations together, and representing by U, the 
whole work u x +u t done by the driving pressure during the 
contact of the teeth, and by S the whole space described by 
the circumference of either pitch circle, we have 

0",= J l + (aa?+bx)aec.(Ti+<p)-\- 

la(l- x y + b(l-x)\ S ec.(>)-9) JURISTS . . . (259) 

by which equation is determined the modulus of two wheels 
driven by involute teeth, when the contact takes place partly 
before and partly behind the line of centres. 

Let the portion of the work U„ which is expended upon 
the friction of the teeth be represented by u. Then 

u= J (ax* -f- bx) sec. (*i +9) + 

\a(l-xy + b{l-x){8ec. (11-9) } HEISTS. 

Now the value of x, which gives to this function its mini- 
mum, and which therefore determines that division of the 
driving arc which corresponds to the greatest economy of 
power, is evidently the value which satisfies the condition 

But differentiating and reducing 



27G THE BEST DIVISION OF THE ANGLE OF CONTACT. 

-y-= | 2ax\$ec. (*)+<?) + sec. (r t — <p)j + 
5 {sec. (i + <p)— sec. (*i — <p){ — 2a sec. (*)— 9) [ U f ; 

-^=2a{sec.(*j+9) + sec. (>]--<p)}TJ 9 : 

Whence it appears that the second condition is always satis- 
fied, and that the first condition is satisfied by that value of 
a?, which is determined by the equation 

2#a?jsec. (1 -f- 9) + sec. (1—9)} + 2>|sec. (11 + 9)— sec. (*j— 9)} — 

2a sec. (*i— p)=0 ; 
Whence we obtain by transposition and reduction 

a?=- J 1— (1H — J tan. *i tan. <p I . 

So that the condition of the greatest economy of power is 
satisfied in respect to involute teeth, when the teeth first 
come into contact before the line of centres at a point whose 
angular distance from it is less than one half the angle sub- 
tended by the pitch by that fractional part of the last-men- 
tioned angle, which is represented by the formula -JI1+- I 

tan. 77 tan. 9, or substituting for b and a their values by the 
formula 



L1P1 • ■ Wi • 

— — sm. p + — ^sm. 9, 

/1 1\ 

cos. 7/ sm. <p 



w, nj 



tan. 7] tan. 9 . . . (260). 



That division of the angle of contact of any two teeth by 
the line of centres, which is consistent with the greatest 
economy of power, is always, therefore, an unequal division, 
the less portion being that which lies before the line of cen- 
tres ; and its fractional defect from one half the angle of con- 
tact, as also the fractional excess of the greater portion above 
one half that angle, is in every case represented by the above 
formula, and is therefore dependent upon the dimensions of 
the wheels, the forms and numbers of the teeth, and the cir- 
cumstances under which the driving and working pressures 
are applied to them.* 

* The division of the arc of contact which corresponds to the greatest eco- 
nomy of power in epicyeloidal teeth, may be determined by precisely the same 
steps. 




THE MODULUS OF A SYSTEM OF TWO WHEELS. 277 



224. The modulus of a system of two wheels dbtven by 
epicycloidal teeth. 

The locus of the point of contact P of any two such teeth 
is evidently the generating circle APH of 
the epicycloidal face of one of the teeth, and 
the hypocycloidal flank of the other (Art. 
202.) ; for it has been shown (Art. 199.), 
that if the pitch circles of the wheel and the 
generating circle APH of the teeth be con- 
ceived to revolve about fixed centres B, C, 
D by their mutual contact at A, then will a 
point P in the circumference of the last-men- 
tioned circle move at the same time upon 
the surfaces of both the teeth which are in 
contact, and therefore always coincide with their point of 
contact, so that the distance AP of the point of contact P of 
the teeth from A, which distance is represented in equation 
(250) by X, is in this case the chord of the arc AP, which 
the generating circle, if it revolved by its contact with 
the pitch circles, would have described, whilst each of the 
pitch circles revolved through a certain angle measured 
from the line of centres. Let the angle which the driven 
wheel (whose centre is C) describes between the period 
when the point of contact P of the teeth passes the line of 
centres, and that when it reaches the position shown in the 
figure be represented as before by +, the arc of the pitch 
circle of that wheel which passes over the point A during that 
period will then be represented by r^. Sow the generating 
circle APH having revolved in contact with this pitch circle, 
an equal arc of that circle will have passed over the point A ; 
whence it follows that the arc AP^r^ ; and that if the radius 
of the generating circle be represented by r, then the angle 

ADP subtended by the arc AP is represented by — ^, or 

T 

by 2<4', if 2e be taken to represent the ratio — of the radius 

of the pitch circle of the driven wheel to the radius of the 
generating circle. Now the chord AP=2AD sin. £ ADP ; 

therefore "k=2r sin. e\—— sin. e\. Substituting this value 

e 

of X in equation (250) ; observing, moreover, that the angle 



278 THE MODULUS OF A SYSTEM OF TWO WHEELS 



PAD represented by & in that equation is equal to-— -J 

2 

it 
ADP, or to -— e\, and that the whole angle \ through 

which the driven wheel is made to revolve by the contact of 

each of its teeth is represented by— , we have 

2tt_ 

U^Y^f \ 1 + {-(- + -) sin. 9 Bin. 4 + — sin. 9l + 
o 

^ sin.?,! sec. (4-9) i <ty+NS ; 
or, assuming L, and L a to remain constant during the cuu 



unng tne cv±i- 
tact of any two teeth representing the constant 1 + — — sin^ + 

— ^sin. 9, by A, and observing that — =— , 

« a »a 

JJ,=F^ \ A /sec. (4— 9)^+ + -(l +— ) si n - <?>/ sin.^i sec. 



(4-9)^ | +NS. 
Now the general integral, / sec. (4— p)^? or 
- / sec. (&^—4>)d{e-^—(p) being represented* by the function 

-y sec. (^-0)^-0) =- J cQs {ei) _^ =J cos .^_ 0) = 

1 /* CQ8. (eip—<p)d{erp—<p) 1 reos.(eiJj —<j>)d{ei>—<l>) 

7 J 1— sin. a («V— 0) ~2e J 1+sin. (eV»— *) + 
1 ncn.W-MM-*) = I, | 1+ S in. («*-♦> | -J log. 

l_ Bn . (^-,) J - 7 log. £ | ^.^.^ J " 



HAVING EPICYCLOIDAL TEETH. 279 

-log. tan. | -r+iW'— 0) [ > its definite integral between the 

2tf 
limits and — has for its expression, 

ij taD -li+l(4»i 

tau - (l-l) 

Also/ sec.(*4— 9) sin.e+^=/ sec.^— 9) sin. {(^+—9) +9} ^ 
o o 

27T 

=r/sec. (^4 / — 9) { 8m - (^+—9) cos. 9-f cos. (^+—9) sin.<p}<#+ 


27T 
/*% 

= / {cos. 9 tan. {e\— 9)+ sm. 9}^ 


=-cos. 9/ tan. (<?+— 9) ^ (0+— 9)H sin. 9. 



Now the general integral/ tan. (^+—9) ^ (^+— 9) has for 
its expression— log. g cos. (e+— 9).* Taking its definite inte- 
gral between the limits and — , we have, therefore, 

n, 

*Z / 2e« \ 

/ih 1 cos. I ~9J 2^ 

sec. (0+— 9) sin. e\d\>— — cos.9log. 



cos. 9 n i 



H — sin.9. 



, 2 sin. j ^+£ (<^-0) [cos. J J-^-^J- \ k 
-log. 



* e I 2 sin. J ^_*( ei /,-0) I cos. J ^(eV-^) [ [ 
llog. £ tan.|^+i( e V-0) |+C. 

• Aan. (et-Qd. je1>-*)= f^ (<*-*) *W~*\ 
«/ J cos. (eip—Q) 

/-dcos. («V-0) . / , A x 

COS. («*-*) =- l0 S- £ C0S -(^-^ 



280 THE MODULUS OF A SYSTEM OF TWO WHEEL8 

Substituting these expressions in the modulus, representing 
j— | D J <P'» and observing that if U 9 represent the work 
yielded by the driven wheel during the action of each tooth, 

then P a a a .— =U a , so that P a a a =^E?, we have 

n 3 2* 

n f ( tan.^+9') , / w v j CO s.<p 

H|lZ!)-^sin. 9 [ j U a + NS . . . (261). 

COS. 9 ^ ' ' 

XT , cos.( — — 9) . ( . , 26* ) 26* 

iNow log. \ w * / = lo2r. « 1+tan. — tan.<p }• cos. — = 

6 £ cos. 9 £ < *■ ' .* 

26* ( 26* ) . 26* 
log. cos. +loff. < 1 -f tan. tan. 9 Y = log. cos. h 

tan. tan. 9— i tan. 2 . tan. a 9 + &c. Substituting this 

expression in the preceding equation, and neglecting terms 
above the first dimension in tan. 9 and sin. 9, 

tan. (?+*>) . , . 

U,= p- A log. ;*>, ' -1 1 + -Mam. 2, log. 

1 26* ( & £ tan. 9' 2e\ nj & « 

cos. — iu a + NS (262). 

225. If the radius r of the generating circle be equal to 
one half the radius r a of the pitch circle of the driven wheel, 
according to the method generally adopted by mechanics 

(Art. 203.), then e=i^ = i—=l. 

In this case, therefore — that is, where the flanks of the 
driven wheel are straight (Art. 210.) — the modulus becomee 

tan. (— + 9') . 

U, =£ Alog. , K , ' -i (l + n A sin. 2 9 log. cos. 2 - [ 
2* ( to £ tan. 9 \ ^7 & * w, J 

U, + NS (263). 



HAYING EPICYCLOIDAL TEETH. 281 

226. Substituting (in equation 262.) for <p' its value -, 



°^' e tan. <p' °£'e (it <p\ 

"Mi -2) 



tan. 



l + tan.(^-?) 1-tan, 9 

\n t 2/ 



\n % 2/ , 2 rt /«* <P\ . 

^^t-^) 1+taD -2 

2tan.| + |tan. 8 (^_|)+|tan. 9 |+&c. 

If, therefore, we assume the teeth in the driven wheel tc 
be so numerous, or n 9 to be so great a number, that the third 

power and all higher powers of tan. I -\ may be ne- 
glected as compared with its first power, and if we neglect 
powers of tan. - above the second, 



tan 
log- s — 



/ , =2 j tan. ( £)+ tan. 5 f ; 

tan. 9 ( W 2 2/ 2 ) ' 



which expression becomes — if we suppose the two arcs 

which enter into it to be so small as to equal their respec- 
tive tangents. 

Again, log.g cos. — = — $1 — I very nearly.* 

* For assume log.e cos. x=aiX^-\-a^x*-\-a i x fi -\- . . . . ; then differentiating, 

—tan. z=2a 1 x-\-4 : a- i x 3 -\-6a2X b -{- ; 

2 
but (Miller, Diff. Cal. p. 95.)— tan. x=— x— ix 8 —- r— rz 5 — . ...; equating, 

3 . 5 

therefore, the co-efficients of these identical series, we have 
1 1 2 

.'.log.e cos. «= — — ■ 



2 3.4 3.5.6 



282 THE MODULUS OF THE RACK AND PINION. 

Substituting these values in equation (262), and perform- 
ing actual multiplication by the factor -=-*-, we have 

U,= { A+4* ^+i) sin. 2? | U,+NS ; 

or substituting for A its value ; and assuming £ sin. 2p= 
sin. <p, since <p is exceedingly small, 

U- | (l + ^ sin. 9l + — sin. 9 ,\ + 

^^^(sim^II^S (264), 

which is the modulus of a wheel and pinion having epicy- 
cloidal teeth, the number of teeth ?i 2 in the driven wheel 
being considerable (see equation 252). 

It is evident that the value of JJ l in the modulus (equa- 
tion 261), admits of a minimum in respect to the value of «/ 
there is, therefore, a given relation of the radius of the 
generating circle of the driving, to that of the driven wheel, 
which relation being observed in striking the epicycloidal 
faces and the hypocycloidal flanks of the teeth of two wheelL 
destined to work with one another, those wheels will work 
with a greater economy of power than they would under any 
other epicycloidal forms of their teeth. This value of e may 
be determined by assuming the differential co-efficient of the 
co-efficient of TJ 3 in equation (261) equal to zero, and solving 
the resulting transcendental equation by the method of 
approximation. 



227. The modulus of the rack and ptnion. 

If the radius r 9 of the pitch circle of the driven wheel be 
supposed infinite (Art. 213.), that wheel becomes a rack, and 
the radius r, of the driving wheel remaining of finite dimen- 
sions, the two constitute a rack and pinion. To determine 
the modulus of the rack and pinion in the case of teeth of 
any form, the number upon the pinion being great, or in 
the case of involute teeth and epicycloidal teeth of any 
number and dimensions, we have only to give to r, an 
infinite value in the moduli already determined in respect 



THE MODULUS OF THE RACK AND PINION. 283 

to these several conditions. But it is to be observed in 
respect to epicycloidal teeth, that n^ becomes infinite with 

T 

r„ whilst the ratio— remains finite, and retains its equality 

T 6 T T 6 

to the ratio — (equation 227), so that— = •£• — =% — =— ; 

if we represent the ratio — by 2^. Making n % and r a infinite 
in each of the equations (252), (254), and (261), and sub- 
stituting — for — in equation (262) ; we have 

1. For the modulus of the rack and pinion when the teeth 
are very small, whatever may be their forms, provided that 
they work truly. 

TJ l= 1 1 + ^sin. Pi+^sin. <p j U.+NS (265). 

2. For the modulus of a rack and pinion, with involute 
teeth of any dimensions (see Jig. 1. p. 255), 

1^= j 1+ (— cos. 7] sin. cp -\ — —sin. ^j sec. (rj— cf>) f TJ 3 + 

NS . . (266). 

3. For the modulus of the rack and pinion, with cycloidal 
and epicycloidal teeth respectively (equation 261), 

' 2e,f ( \ ap x r 7 ° E tan. <p 

~^r l °s- -^- f u -+ NS - 

In each of which cases the value of N" is determined by 
making r* infinite in equation (247). 



e ei e w 2 

Tla Til «l Wj 



e is infinite. The friction of the rack upon its guides is not taken into account 
in the above equations. 



284 



CONICAL WHEELS. 



Conical or Bevil Wheels. 




a«t- 



228. These wheels are used to communicate a motion of 
rotation to any given axis from another, inclined to the first 
at any angle. 

Let AF be an axis to which a motion of rotation is to be 
communicated from another axis AE 
inclined to the first at any angle EAF, 
by means of bevil wheels. 

Divide the angle EAF by the straight 
line AD, so that DO and DX, perpen- 
diculars from any point D in AD upon 
AE and AF respectively, may be to 
one another as the numbers of teeth 
which it is required to place upon the 
two wheels.* 

Suppose a cone to be generated by the revolution of the 
line AD about AE, and another by the revolution of the 
line AD about AF. Then if these cones were made to 
revolve in contact about the fixed axes AE and AF, their 
surfaces would roll upon one another along their whole line 
of contact DA, so that no part of the surface of one would 
slide upon that of the other, and thus the whole surface 
of the one cone, which passes in a given time over the line 
of contact AD, be equal to the whole surface of the other, 
which passes over that line in the same time. For it is 
evident that if n x times the circumference of the circle DP 
be equal to n % times that of the circle DI and these circles 
be conceived to revolve in contact carrying the cones with 
them, whilst the cone DAP makes n x revolutions, the cone 



* This division of the angle EAF may be made as follows : — Draw ST and 
UW from any points S and U in the straight lines AE and AF at right angles 
to those lines respectively, and having their 
lengths in the ratio of the numbers of . teeth 
which it is required to place upon the two wheels ; 
and through the extremities T and W of these 
lines draw TD and WD parallel to AE and AF 
respectively, and meeting in D. A straight line 
drawn from A through D will then make the 
required division of the angle ; for if DO and 
DN be drawn perpendicular to AE and AF, they 
will evidently be equal to UW and ST, and there- 
fore in the required proportion of the numbers 
of the teeth ; moreover, any other two lines 
drawn perpendicular to AE and AF from any 
other point in AD will manifestly be in the same proportion as DO and DN. 




CONICAL WHEELS. 285 

DAI will make n^ revolutions; so that whilst any other 
circle GH of the one cone makes n x revolutions, the corre- 
sponding circle HK of the other cone will make n 2 revolu- 
tions : but n 1 times the circumference of the circle GH 
is equal to n^ times that of the circle HK, for the diameters 
of these circles, and therefore their circumferences, are to 
one another (by similar triangles) in the same proportion as 
the diameters and the circumferences of the circles DP and 
DI. Since, then, whilst the cones make n x and n^ revolutions 
respectively, the circles HG and HK are carried through. n x 
and n 2 revolutions respectively, and that n x times the circum- 
ference of HG is equal to ti 2 times that of HK, therefore 
the circles HG and HK roll in contact through the whole of 
that space, nowhere sliding upon one another. And the 
same is true of any other corresponding circles on the cones ; 
whence it follows that their whole surfaces are made to roll 
upon one another by their mutual contact, no two parts 
being made to slide upon one another by the rolling of the 
rest. 

The rotation of the one axis might therefore be communi- 
cated to the other by the rolling of two such cones in con- 
tact, the surface of the one cone carrying with it the surface 
of the other, along the line of contact AD, by reason of the 
mutual friction of their surfaces, supposing that they could 
be so pressed upon one another as to produce a friction equal 
to the pressure under which the motion is communicated, or 
the work transferred. In such a case, the angular velocities 
of the two axes would evidently be to one another (equation 
227) inversely, as the circumferences of any two correspond- 
ing circles DP and DI upon the cones, or inversely as their 
radii ND and OD, that is (by construction) inversely as the 
numbers and teeth which it is supposed to cut upon the 
wheels. 

When, however, any considerable pressure accompanies 
the motion to be communicated, the friction of two such 
cones becomes insufficient, and it becomes necessary to 
transfer it by the intervention of bevil teeth. It is the cha- 
racteristic property of these teeth that they cause the motion 
to be transferred by their successive contact, precisely as it 
would by the continued contact of the surfaces of the 
cones. 



286 



CONICAL WHEELS. 



229. To describe the teeth of bevil wheels* 

From D let FDE be wrawn at right angles to AD, inter- 
secting the axes AE and AF of the two cones in E and F ; 
suppose conical surfaces to be generated by the revolution 
of the lines DE and DF about AE and AF respectively ; 




and let these conical surfaces be truncated by planes LM 
and XY respectively perpendicular to their axes AE and 
AF, leaving the distances DL and DY about equal to the 
depths which it is proposed to assign to the teeth. Let now 
the conical surface LDPM be conceived to be developed 
upon a plane perpendicular to AD, and passing through the 
point D, and let the conical surface XIDY be in like 
manner developed, and upon the same plane. When thus 
developed, these conical surfaces will have be- 
come the plane surfaces of two segmental annul i 
~MJ*_pm and IXxif, whose centres are in the 
points E and F of the axes AE and AF, and 
which touch one another in the point D of the 
line of contact AD of the cones. 

Let now plane or spur teeth be struck upon 
the circles rp and K, such as would cause them 

* The method here given appears first to have been published by Mr. Tred- 
gold in his edition of Buchanan's Essay on Jfillwork, 1823, p. 103. 

f The lines MP and prn in the development, coincided upon the cone, as also 
the lines IX and ix ; the other letters upon the development in the above 




CONICAL WHEELS. 287 

to drive one another as they would be driven by their 
mutual contact ; that is, let these circles Tj> and \i be taken 
as the pitch circles of snch teeth, and let the teeth be 
described, by any of the methods before explained, so that 
they may drive one another correctly. Let, moreover, their 
pitches be such, that there may be placed as many such 
teeth on the circumference Pj? as there are to be teeth 
upon the bevil wheel HP, and as many on \i as upon the 
wheel HI. 

Having struck upon a flexible surface as many of the first 
teeth as are necessary to constitute a pattern, apply it to 
the conical surface DLMP, and trace off the teeth from it 
upon that surface, and proceed in the same manner with the 
surface DIXY. 

Take DH equal to the proposed lengths of the teeth, draw 
«/* through H perpendicular to AD, and terminate the wheels 
at their lesser extremities by concave surfaces HGmZ and 
HKoey described in the same way as the convex surfaces 
which form their greater extremities. Proceed, moreover, 
in the construction of pattern teeth precisely in the same 
way in respect to those surfaces as the other; and trace out 
the teeth from these patterns on the lesser extremities as on 
the greater, taking care that any two similar points in the 
teeth traced upon the greater and lesser extremities shall lie 
in the same straight line passing through A. The pattern 
teeth thus traced upon the two extremities of the wheels are 
the extreme boundaries or edges of the teeth to be placed 
upon them, and are a sufficient guide to the workman in 
cutting them. 



230. To prove that teeth thus constructed will work truly 
with one another. 

It is evident that if two exceedingly thin wheels had been 
taken in a plane perpendicular to AD {fig. p. 286.) passing 

figure represent points which are identical with those shown by the same let- 
ters in the preceding figure. In that figure the conical surfaces are shown 
developed, not in a plane perpendicular to AD, but in the plane which contains 
that line and the lines AE and AF, and which is perpendicular to the last-men- 
tioned plane. It is evidently unnecessary, in the construction of the pattern 
teeth, actually to develope the conical extremities of the wheels as above 
described ; we have only to determine the lengths of the radii DE and DF by 
construction, and with them to describe two arcs, Pp, K, for the pitch circles 
of the teeth, and to set off the pitches upon them of the same lengths as the 
pitches upon the circles DP and DI, which last are determined by the numbers 
of teeth required to be cut upon the wheels respectively. 



288 THE MODULUS OF A SYSTEM 

through tin 1 point D, and having their centres in E and F; 
and if teeth bad been cut upon these wheels according tc 
the pattern above described, then would these wheels have 
worked truly with one another, and the ratio of their angu- 
lar velocities have been inversely that of ED to FD, or (by 
similar triangles) inversely that of ND to OD ; which is the 
ratio required to be given to the angular velocities of the 
bevil wheels. 

Now it is evident that that portion of each of the conical 
surfaces DPML and DIXY which is at any instant passing 
through the line LY is at that instant revolving in the plane 
perpendicular to AD which passes through the point D, the 
one surface revolving in that plane about the centre E, and 
the other about the centre F ; those portions of the teeth of 
the bevil wheels which lie in these two conical surfaces will 
therefore drive one another truly, at the instant when they 
are passing through the line LY, if they be cut of the forms 
which they must have had to drive one another truly (and 
with the required ratio of their angular velocities) had they 
acted entirely in the above-mentioned plane perpendicular 
to AD and round the centres E and F. Now this is pre- 
cisely the form in which they have been cut. Those por- 
tions of the bevil teeth which lie in the conical surfaces 
DPML and DIXY will therefore drive one another truly at 
the instant when they pass through the line LY ; and there- 
fore they will drive one another truly through an exceedingly 
small distance on either side of that line. Now it is only 
through an exceedingly small distance on either side of that 
line that any two given teeth remain in contact with one 
another. Thus, then, it follows that those portions of the 
teeth which lie in the conical surfaces DM and DX work 
truly with one another. 

Now conceive the faces of the teeth to be intersected by an 
infinity of conical surfaces parallel and similar to DM and 
DX ; precisely in the same way it may be shown that those 
portions of the teeth which lie in each of this infinite num- 
ber of conical surfaces work truly with one another; whence 
it follows that the whole surfaces of the teeth, constructed as 
above, work truly together. 



231. The modulus of a system of two conical or 
bevil wheels. 

Let the pressure P x and P a be applied to the conical 



OF TWO CONICAL WHEELS. 289 

wheels represented in the accompanying figure at perpen- 
dicular distances a x and # 2 from their axes CB and CG ; let 
the length AF of their teeth be represented by b ; let the 
distance of any point in this line from F be represented by 
a?, and conceive it to be divided into an exceedingly great 
number of equal parts, each represented by A#. Through 
each of these points of division imagine planes to be drawn 



cj^k:. 



perpendicular to the axes CB and CG of the wheels, dividing 
the whole of each wheel into elements or laminse of equal 
thickness ; and let the pressures P 4 and P 2 be conceived to be 
equally distributed to these laminae. The pressure thus dis- 

tributed to each will then be represented by — ± Ax on the 

p 

one wheel, and _ lAx on the other. Let^ and^> 2 represent 

the two pressures thus applied to the extreme laminse AH 
and AK of the wheels, and let them be in equilibrium when 
thus applied to those sections separately and independently 
of the rest ; then if R represent the pressure sustained along 
that narrow portion of the surface of contact of the teeth of 
the wheels which is included within these laminse, and if R, 
and R 2 represent the resolved parts of the pressure R in the 
directions of the planes AH and AK of these laminse, the 
pressures p x and R, applied to the circle AH are pressures 
in equilibrium, as also the pressures p^ and R 2 applied to the 
circle AK. If, therefore, we represent as before (Art. 216.) 
by m, and m. 2j the perpendiculars from B and G upon the 

19 



290 THE MODULUS OF A SYSTEM 

directions of R, and R 2 , and by L, and L 2 , the distances be- 
tween the feet of the perpendiculars a„ m, and a„ m, we 
have (equation 236, 237), neglecting the weights of the 
wheels, 



(267). 



p, and p 2 representing the radii of the axes of the two wheels, 
and 9 X and <p 2 the corresponding limiting angles of resistance. 
Let 7, and / 2 represent the inclinations of the direction of R 
to the planes of AH and AK respectively ; then 

~R J= R cos. y 1? R 2 = R cos. y r 

Now it has been shown in the preceding article, that the 
action of that part of the surface of contact of the teeth which 
is included in each of the laminae AH, AK, is identical with 
the action of teeth of the same form and pitch upon two 
cylindrical wheels AD and AL of the same small thickness, 
situated in a plane EAD perpendicular to AC, and having 
their centres in the intersections, b and g, with that plane of 
the axes CB and CG produced. The reciprocal pressure R 
of the teeth of the element has therefore its direction in the 
plane EAD ; and if its direction coincided with the line of 
centres DL of the two circles EA and AD, then would its 
inclinations to the planes of AH and AK be represented by 
DAH and LAK, or by ACB and ACG. 

The direction of R is however, in every case, inclined to 
the line of centres at a certain angle, which has been shown 
(Art. 216.) to be represented in every position of the teeth, 
after the point of contact has passed the line of centres by 
(d + q>) ; where 6 represents the inclination to AL of the line 
X, which is drawn from the point of contact A of the pitch 
circles to the point of contact of the teeth, and where <p repre- 
sents the limiting angle of resistance between the surfaces of 
the teeth. To determine the inclination y x of RA to the 
plane of the circle AH, its inclination RAD to the line of 
centres being thus represented by (# + p), and the inclination 
of the plane AD, in which it acts, to the plane AH being 
DAH, which is equal to ACB, let this last angle be repre- 




OF TWO CONICAL WHEELS. 291 

sented by i x ; and let Aa in the accom- 
panying figure represent the intersection 
of the planes AD and AH ; Aard repre- 
senting a portion of the former plane and 
Aach of the latter. Let moreover At 
represent the direction of the pressure R 
in the former plane and let Ad and AA be portions of the 
lines AD and AH of the preceding figure. Draw re per- 
pendicular to the plane Aach, and rd and eh parallel to Aa, 
and join dh ; then rAc represents the inclination y x of the 
direction of R to the plane AD, dAr represents the inclina- 
tion (<p-M) of AR to AH, and dAh represents the inclination 
i x of the planes AD and AH to one another. Also, since Aa 
is perpendicular to the plane Ahd, therefore dr is perpen- 
dicular to that plane, 

,\ re = Ar sin. y x = Ad sec. (d 4- <p) sin. /,. 
Also hd = Ad sin. »„ but re = hd, 

.\ Ad sec. (d + <p) sin. y x =Adsm. i,; 
/. sin. y x = cos. (0 + <p) sin. i x . 

In like manner it may be shown that sin. / 2 = cos. (d + p) 
sin. » 2 , i 2 being taken to represent the inclination EAL of the 
planes AE and AK, which angle is also equal to the angle 
ACG. 

From the above equations it follows that 

R 1 =R cos. y x =H 4/ I — cos. a (4 + 9) sin. \ i ^g. 

R 2 =Rcos.7 2 =R|/l-cos. 2 (^+9)sin. a «J V h 

From the centre b of the circle AD draw bm perpendicular 
to RA, then is BM (the perpendicular let tall from the 
centre of the circle AH upon the direction of R,) the projec- 
tion of bm upon the plane of the circle AH. To determine 
the inclination of bm to the plane AH, draw An parallel to 
bm; the sine of the inclination of An to the plane AH is 
then determined to be cos. ~DAn . sin. i„ precisely as the sine 
of the inclination of Am to the same plane was before deter- 
mined to be cos. DAm . sin. i t . 

it tf 

Now DAn =Abm = 5 —DAR = 5 — (6 + <p) ; therefore the 

sine of the inclination of An, and therefore of bm, to the plane 



292 



THE MODULUS OF A SYSTEM 






*$£::„. 




•« 



AH is represented by the formula sin. (0 + 9) sin. »„ and the 
cosine of its inclination by \/l — sin. \6 -f 9) sin. \ ; 

/. m 1 =B^l=bm ^/l — sin. '(0 + 9) sin. \. 

Now it has been shown (Art. 216.) that the perpendicular 
bra let fall from the centre of a spur wheel upon the direc- 
tion of the pressure upon its teeth is, in any position of their 
point of contact, represented (equation 239) by the formula, 

r t sin. (d + 9) + x sin. 9, 

where 0, 9, X represent the same quantities which they have 
been taken to represent in this article ; but r, represents the 
radius bA of the circle AD, instead of the radius BA of the 
circle AH; now 5A=BA sec. DAH=r 1 sec. i t ; substituting 
this value for r 1 in the preceding formula, we have 

bm—r x sin. (d + 9)sec. ij + Xsin. 9; 
:.m l = \r 1 sin. (d + 9) sec. 1,+Xsin. 9} 
¥1— sin. "(4 +9) am. \« 

Similarly it may be shown that [- (269). 

n^ — \r, sin. (d + 9) sec. »,— X sin. 9} 
Vl— sin. '(4 +9) sin. \. 

Substituting the values of m, and m^ above determined, 
and also the values of R x and R a (equations 268) in equations 



OF TWO CONICAL WHEELS. 293 

(267), and eliminating R between those equations, a relation 
will be determined between j> l and jp 2 which is applicable to 
any distance of the point of contact of the teeth from the 
line of centres. 

Let it now be assumed that the number of the teeth of 

the driven wheel is considerable, so that the angle — tra« 

versed by the point of contact of each tooth may be small, 
and the greatest value of the line \ the chord of an exceed- 
ingly small arc of the pitch circle of the driven wheel. In 

this case d+9 will very nearly equal - (Art. 220.); so that 

cos. *(d+<p) will be an exceedingly small quantity and may 
be neglected, and sin. (d+<p) very nearly equal tmity. Sub- 
stituting these values in equations (268) and (269) we have 

3^=11, R 2 =R, 

m 1 =7 , 1 +^ sin. 9 cos. i„ m 2 =r 2 — X sin. <p cos. « 2 . 

Substituting these values in equations (267) and dividing 
those equations by one another so as to eliminate R, 

L, 



. • /PrU\ • 

f.+Xsm. <p cos. i- + sin. <p. 



r a — Xsin. 9 cos. » 2 — y 2 - 1 ) sin. 9 2 

x . /p,L 

1 + 



rPA\ • 
1 H — sin. 9 cos. », + sin. 9. 



-^ 8 1 gm. 9 cos. < 2 — sin. <p„ 

Whence performing actual division by the denominator of 
the fraction, and neglecting terms involving dimensions 
above the first in sin. 9, sin. p l5 sin. <p 2 , 



+ 



7) ar ( /cos.', cos. i„\ . /p,L. \ . 

^i = ^ U +X 2 + 2 sin. 9+ — sin. 9, 

($)*■*!• 

Now if + represent the angle described by the driven 
wheel or circle ELA, whilst any two teeth are in contact, 
since X is very nearJ.y a chord of that circle subtending this 
small angle + (Art. 220.) ; .*. X = r^. Let * represent the 



294 THE MODULUS OF A SYSTEM. 

angle described by the conical wheel FK, whilst the circk 
E1A describes the angle + ; then, since the pitch circle of 
the thin wheel AK and the circle ELA revolve in contact at 
A, they describe equal arcs whilst they thus revolve, respec- 
tively, through the unequal angles + and *. Moreover, the 

radius A.g of the circle AL=AG sec. GAg=r t sec. «„ there- 
fore 4^, sec. i a =$r a ; 

.•.4,=* cos. » a (270). 

Substituting the above valves of + and X, and observing 

r x n x 
Vx a>i n x ( -. /cos. 1. cos. l\ , 

g) S in. ?1+ g) 8 in.,,| (271). 

Multiplying both sides of this equation by^>, -J_?, and ob- 

7l> 71 && 7b 

serving that^a, — =^p l a l a , and that —A* is the exceed- 
ingly small angle described by the driving wheel AN, whilst 
the driven wheel describes the angle a* 5 so that if Aw, repre- 
sent the work done by the pressure^ upon the lamina AH, 
whilst the angle a* is described by the driven wheel, then 

ya — a^^a-m , we have 

ex 7l t 

A^ _ /COS.'. COS. L\ 

^*=iv*. 1 1 + ". \^~ + "^T/ * cos - '■ sm - 9 + 

(£)sin. ?1+ (g)si,^ 

or assuming A* infinitely small, and integrating between the 
limits and — (Art. 220.), 

%*P*a* \ + /cos. 1. cos. i A 
u.= -^ 2 - 3 < 1 +* -+ -cos. i a sin. 9 + 

(PiL\sin. 9l + (^)sin. <pj. 
\a 1 7' 1 / \ajrj ) 



OF TWO CONICAL WHEELS. 295 

Now the above relation between the work u t done by the 
pressure p 1 upon the extreme element AH of the driving 
wheel whilst any two teeth are in contact, and the pressure 
j?^ opposed to the motion of the corresponding element of 
the driven wheel, is evidently applicable to any other two 
corresponding elements ; the values of js? 2 , r 1? r„ L x and L a 

E roper to those elements being substituted in the formula. 
t, therefore, we represent by aU, that increment of the 
whole work U, done upon the driving wheel, which is due 
to any one of the elements into which we have imagined 
that wheel to be divided, and if we substitute for ^? a its 

P 
value -t^ a #, assign to L 1? L a , r„ r % their values proper to that 

element, and represent those values by L, I/, r, r\ 
TT 2*-P 2 & 2 ( /cos. « x cos. » 2 \ 

or assuming &x infinitely small, and integrating between the 
limits and b, and observing that P 2 & 2 — represents the 

whole work U 2 done upon the driven wheel under the con- 
stant pressure P 3 during the contact of any two teeth, 

TT TT /cos. i, cos. i 2 \ 

U^IT.Jl +*!—— 1 +.—J| cos. i a am. 9+ 

b b 

.. cp. /*L , p„ sin. <p_ fU , , 

rJr^b^rJ^ (272) - 

1 A 3 A * 



z>&. 



Now &+# being taken to represent the distance of the 
point of contact of any two such elements from C, and a to 
represent the distance CF, the radii r and r' of these ele- 
ments are evidently (by similar triangles) represented by 

a + x I x\ a + x I x\ 

—r x or \ 1 + - j r„ and -^-r« or 1 1 + - j r„ r x and r 2 repre- 
senting the radii of the extreme elements NF and OF, or of 
the pitch circles of the lesser extremities of the wheels. 
Also assuming, as we have done, the pressures H^ and R, 



296 



THE MODI I. US OF A SYSTEM 




to be perpendicular to the lines BA, 
GA joining the centre of each ele- 
ment with their point of contact A, 
so that the points M and N (see jig. 
p. 292.) coincide with the point A 
(see accompanying figure)* ; and re- 
presenting the angles ABD and ACE 
made by the perpendiculars DB and 
CE with the line of centres by d, and 

^respectively; observing also that AD 2 =B A 2 — 2BA. BD 
cos. ABD + ED*, so that (^) =1-2 (|^) cos. ABD + 

/ ) , we have, substituting, in the second number of this 

equation, for BA or r its value r x 1 1 -f - j 

or expanding the binomials in this expression, observing 
that - is an exceedingly small quantity, neglecting terms 

Ob 

involving powers of that quantity above the first, and 
reducing, 

■G)(«**-ge ■"> 

Now L 1 representing the value of L when a?=0, and & re 
maining constant, 



L, 



ft) - — ■©-*+©' 

" \rj\ J rj~ \rj \rj' 



* The circles in this figure represent two of the corresponding laminae intc 
which wheels have been imagined to be divided; they are not, therefore, in the 
same plane. Their planes intersect in AH. 



OF TWO CONICAL WHEELS. 297 

Let now the angle ADB, made in respect to the first ele- 
ment of the driving wheel between the perpendicular BD or 
a y and the chord AD or L, be represented by 7/ and let r\ % 
represent the corresponding angle in the driven wheel, then 

/L \ a 
L;- 2La cos. % + a? = r x % ;. (— ' j - 

L^ jay . 

2 ? ~.i,+ y =i; 

Substituting these values of I— J and 2 J— ) I cos. 6 — — \ 
in equation (273) ; 



Extracting the square root of the binomial, and neglecting 

terms involving powers of - above the first, 

a 

L L, laA lx\ aAJj x x ) 

— = I— I I- ) cos. % — — i cos. v. \ ; 

m . ~w~n Pi sin. <p. /*L _ p. sin. <p. f L. J ) 

•• CEq^on 272) t^ -^^ ^ -* -cos.,, . 

1 

P,sin.9 3 /*L' p 3 sin.<p Q ( L 2 5 ) 

Similarly -^—J -, dx = — — | - - * - cos. % J . 

Substituting these values in the modulus (equation 272), 

TT XT i „ / cos - 'i cos - 'a\ 

TJ l = U 2 ] l+*rl i + -lcos. < 9 Bin. q> + 

P.Bin.^/L, 5 \ p 2 sin.<p,/L, & \ ) 

— l — — — \~ ios. ^ H 1 J- COS. 77., . \ 



298 



THE MODULUS OF A SYSTEM 



Now let the angle BCG, or the inclination of the axes, 
from one to the other of which motion is transferred by the 
wheels, be represented by % ; therefore », + * 3 =2<. Also a 
6in. i l =r l and a sin. ' a =r a , 



sin. », 
sin. i. 



n. 



sin. \ __ sin. \ 1 



n. 



a J 



»; 



cos. \ 

~^7~ 



COS. \ 



71 



. 1 1 cos. 3 *^ ggs^ /COS. <, COS. u /cos.*, cos.a 

(cos. I, COS. l 9 \ /l COS. I, 1\ 

" \~ + ~^T/ K c^s77 3 ~ »J cos ' '■ 

. /cos. ', COS. « 3 \ 



cos. < 3 = 



1 COS. 



Now 



COS 



. «, _ cos, {' + £(',— Qi 1— tan. •£(<,— < 3 )tan. i 



also 



cos. « 3 cos. \\— £(',— ' a )} 1 + tan. £(«,— » 3 )tan. » ' 

,_ sin. « x _ sin, j'+j-^— Qj __ tan, i + tan. \ (»,— »,) 
ti 3 ~ sin. i 3 ~~ sin. {'— J('i~Of ~~ taru ' ~~ tan - i('x _ '0 



n 



cos. « 



tan. -| (»,— « 3 ) 



71,— n 3 , 

1 — — * tan. '« 

n. + n, 



n,—n 



? tan. » ; 



cos.., ^ tan _,, 

7l,+7l 3 



7i, + 7i a 

_ (ti, + 7i 3 ) — K — 7i,) tan. a i . 
~(^+7i 3 ) + (n 1 — nJtan.S ' 



1 cos. »,1 1 / COS. I. \ 

71, COS. < 3 7l a n,7i 3 \ a cos. » a 7 

—1 K'-Q + «— Qtan. a i _ 
71,71, (7i 1 + 7i a )H-(7i 1 — 7t 3 )tan. 2 < 



OF TWO CONICAL WHEELS. 299 



(h - B sec - a ' b ~ h) 



(1+1) _ (1^1) ^ ■, (1+1 ) cos ,,_ (1_I) 6in ,, ' 

/ COS. « x COS. < 2 \ / 1 1 \ 

(l-l)sin.% = (l + l)-l^. 

Substituting in the preceding relation, between U x and U a , 
Uj = 1 1+* 1 (— + — ) f sin. 9 + - 

which is the modulus of the conical or bevil wheel, neglecting 
the influence of the weight of the wheel. 

If for cos. % and cos. % we substitute their values (see 
p. 297), we shall obtain by reduction 

tt i i , i I 1 1 \ 2sin. a <{ . Min.9, j _ 

11,= ] 1+*1 (— + — ) f sm.9 + - -1 L, 

I I \n x nj n x J r ^ a x r x ( l 

^l}" w, • 

from which equation it is manifest that the most favourable 
directions of the driving or working pressures are those 
determined by the equations 



232. It is evident, that if the plane of the revolution 
of such a wheel be vertical, the influence of its weight must 
be very nearly the same as that of a cylindrical or spur 



300 THE MODULUS OF A SYSTEM OF TWO CONICAL WHEELS. 

wheel of the same weight, having a radius equal to the mean 
radius of the conical wheel, and revolving also in a vertical 
plane. If the axis of the wheel be not horizontal, its weight 
must be resolved into two pressures, one acting in the plane 
of the wheel, and the other at right angles to it; the latter is 
effective only on the extremity of the axis, where it is borne 
as by a pivot, so that the work expended by reason of it may 
be determined by Art. 176, and will be found to present 
itself under the form of ]ST 2 . S, where N 2 is a constant and S 
the space described by the pitch circle of the wheel, whilst 
the work t^ is done. The resolved weight in the plane 
of the wheel must be substituted for the weight of the wheel 
in equation (247), which determines the value of N". Assum- 
ing the value of N, this substitution being made, to be repre- 
sented by N l5 the whole of the second term of the modulus 
will thus present itself under the form (N\ + !N" a )S. 

i j/1 1\ 2sin. 2 <) . PxSin.^ 
: . TJ.= 1 1 + * 1 — + — 1 — f sin. <p + ~ 

tli. ' ft \ p 2 sin.<p 2 /L 2 ft \| _. 

(N" x +:N" a )S (276). 

233. Comparing the modulus of a system of two conical 
wheels with that of a system of two cylindrical wheels 
(equation 252), it will be seen that the fractional excess 
of the work tJ 2 lost by the friction of the latter over that 
lost by the friction of the former is represented by the 
formula 

2* sin. 2 » sin, <p J/p. . , p 2 \ 

+ —+ i~ ~ cos - ^i sm - *» + 7 cos. iksiii.<p J • • . 

The first term of this expression is due to the friction of 
the teeth of the wheels alone, as distinguished from the fric- 
tion of their axes ; the latter is due exclusively to the friction 
of the axes. Both terms are essentially positive, since r\ x 

and 7/ 2 are in every case less than -. 

Thus, then, it appears that the loss of power due to the 
friction of bevil wheels is (other things being the same) 
essentially less than that due to the friction of spur wheels, 
so that there is an economy of power in the substitution of 




THE MODULUS OF A TRAIN OF WHEELS. 301 

a bevil for a spur wheel wherever such substitution is prac- 
ticable. This result is entirety consistent with the experience 
of engineers, to whom it is well known that bevil wheels run 
liqhter than spur wheels. 



234. The Modulus of a Train of Wheels. 

In a train of wheels such as that shown in the accompany- 
ing figure, let the radii of their 
pitch circles be represented in 
order by r„ r„ r , . . . r 4 , begin- 
ning from the driving wheel ; 
and let a x represent the perpen- 
dicular distance of the driving 
pressure from the centre of that 
wheel, and & 2 that of the driven 
pressure or resistance from the centre of the last wheel of the 
train ; U 1 the work done upon the first wheel, u^ the work 
yielded by the second wheel to the third, u^ that yielded by 
the fourth to the fifth, &c, and U 2 the work yielded by the 
last or n th wheel upon the resistance, then is the relation be- 
tween Uj and u^ determined by the modulus (equation 252), 
it being observed that the point of application of the resist- 
ance on the third wheel is its point of contact b with the 
third wheel, so that in this case & 3 =r s . 

These substitutions being made, and L 2 being taken to 
represent the distance between the point b and the projection 
of the point a upon the third wheel, we have 

u HM^) sin - 9+ ^ sin - 9 ' + 

To determine, in like manner, the relation between w a and 
w 8 , or the modulus of the third and fourth wheels, let it be 
observed that the work u 7 which drives the third wheel lias 
been considered to be done upon it at its point of contact b 
with the fourth ; so that in this case the distance between the 
point of contact of the driving and driven wheels and the 
foot of the perpendicular let fall upon the driving pressure 
from the centre of the driving wheel vanishes* and the term 

* See note p. 266. 



THE MODULUS OF A TRAIN OF WIFEELS. 

which involves the value of L, representing that line disap- 
pears from the modulus, whilst the perpendicular upon the 
driving pressure from the centre of the driving wheel be- 
comes 7%. Let it also be observed, that the work of the 
fourth wheel is done at the point of contact c of the fifth and 
sixth wheels, so that the perpendicular upon the direction of 
that work from the axis of the driven wheel is r b . AVe shall 
thus obtain for the modulus of the third and fourth wheels. 



UJ= 



1 H^7) sin -* + ^ siQ -M u '+ K ' S '- 



In which expression L 3 represents the distance between the 
point c and the projection of the point b upon the fifth 
wheel. 

In like manner it may be shown, that the modulus of the 
fifth and sixth wheels, or the relation between u 3 and u t , is 

u 3 = | l + *(_ + i)sin.<p + -^sin. <p 4 } t* 4 +N, . S, ; 
( \n 6 nj r t r, ) 

and that of the seventh and eighth wheels, or the relation 
between u A and u b , 



u A 



{Mrt) sin - ?+ ^ siii -*>l «.+*.• s - 



and that, if the whole number of wheels be represented by 
2/?. or the number of pairs of wheels in the train by^?, then 
is the modulus of the last pair, 



u p — X 1 + W j Jsin. <p + 

( \n 2 p-i nip! 

Ijp+ipp+i^ 






In which expressions the symbols 2s\, N„ N, . . . Np, are 
taken to represent, in respect to the successive pairs of wheels 
of the train, the values of that function (equation 2-iT), 
which determines the friction due to the weights of those 
wheels ; and each of the symbols L„ L 3 , L 4 . . . Lp , the dis- 
tance between the point of contact of a corresponding pair 
of wheels and the projection upon its plane of the point of 
contact of the next preceding pair in the train ; whilst the 
symbols n l5 t? 3 , n t . . . n, p , represent the numbers of teeth in 
the wheels ; r„ r„ r s , . . . r„ p , the radii of their pitch circles ; 
and S 1? S„ S, S p , the spaces described by their points ot 



THE MODULUS OF A TRAIN OF WHEELS. 303 

contact a, J, e, &c. whilst the work U, is done upon the first 
wheel of the train. 

Let us suppose the co-efficients of u^ u 3 , u 4 . . . U 2 , in these 
moduli to be represented by (1+^)? (l+^ 2 )> (l-hf*.) • - • • 
(l-l-uo) ; they will then become 

«,=(i+fs)^+isr,..s iJ 

w,=(l +f*,) w 4 +N, . s„ 
&c.=&c. 

Eliminating u„ u 3 , u 4 . . . u p , between these equations, we 
shall obtain an equation of the form 

TJ^I+mO (1 +H0 (1+f*,) . . - (l+ffcpJU.+N . S . . . (277), 
where 

+ (1+^(1+^) .... (l+.^iJHp.Sp (278). 

Now let it be observed, that the space described by the first 
wheel, at distance unity from its centre, whilst the space S, 

is described by its circumference, is represented by — , and 

that this same space is represented by — if S represent the 

space described in the same time by the foot of the per- 
pendicular a x , or the space through which the moving 
pressure may be conceived to work during that time; so 

S S 
that — = . Also let it be observed that the space de- 

scribed by the third wheel, at distance unity from its centre, 

is the same with that described at the same distance from 

« S S 
its centre by the second wheel, so that — =— ; in like 

manner that the spaces described at distances unity from 
their centres by the fourth and fifth wheels are the same, so 

s s s s 

that — =— ; and similarly, that — =— , &c. = &c. ; and 
finally, A_ = V?_. 

?V-1 7*2j>-9 



804 tiii: MODULUS of a tkaix of WHEELS. 

Multiplying the two first of these equations together, then 
the three first, the/ow first, &c, and transposing, we have 

a, a x • r % \aj \nj ' t 

g __ r, . r, . r t |g __ /t\\ fejJK\g 
1 a x .r % . r A \aj U, . nj ' 
g r,.r,.r..r, s= /r,| /n, ■ „, ■ „,| 

&C.=&C 

a, . r, . t\ . . . /'op-2 \a 1 / W 3 . n 4 . . . n*^/ 

Substituting these values of S„ S 9 , &c. in equation (278), 
and dividing by S, we have 

or if we observe that the quantities f*,, /x 5 , ja 3 , are composed 
of terms all of which are of one dimension in sin. p, sin. <p,, 
sin. p s , (fee. and that the quantities N„ N„ N w &c. (equation 
24:7 i are all likewise of one dimension in those exceedingly 
small quantities ; and if we neglect terms above the nxst 
dimension in those quantities, then 

ess)"* •••(•••<«»> 

If in like manner we neglect in equation (277) terms of 
more than one dimension in m-„ fx 2 , ja,, &c. we have 

U l ={l+f* 1 +»s+is+ • . • +MU.+N . S. 

Now ft.= * ( _ « —] sin. <p + _^i sin. 9, + — — sin. <?„ 
\n t n,7 a,^ r,r % 

»,=* ( l _ ) sin. 5 + -^ sin. © l? 



THE MODTJLTTS OF A TRAIN OF WHEELS. 305 

M 3 = * ( j ) sin. 9 + — *h sin. <p 4 , 

\n 6 nj r a r, 

&c.=&c. 

^ = * (JL + -L) sin. 9 + ^±^ sin., +1 . 

Substituting these values of f*,, f* a , &c. in the preceding 
equation, 

.-. U - {l + * (1+1+ JL L\ sin.9+ hh sin.9,4- 

hh s in. 9 2 + ^sin.9 3 +... Ly+lP ^ sin. J?+ i}H 3 +KS . (280), 



which is a general expression for the modulus of a train of 
any number of wheels. 



235. The work TJ 1 which must be done upon the first 
wheel of a train to yield a given amount U a at the last wheel, 
exceeds the work TJ 9 , or, in other words, the work done upon 
the driving point exceeds that yielded at the working point, 
by a quantity which is represented by the expression 

« — -f— + ... H sin. 9.U a + (- L11 sin. 9,+ -^ sin. (p, 

+ .... + ^±f^sin. 1)+1 )u s+ NS. . .(281). 

In which expression the first term represents the expenditure 
of work due to the friction of the teeth, and varies directly as 
the work U a , which is done by the machine. The second 
term represents the expenditure of work due to the friction 
of the axes of the wheels, and varies in like manner directly 
as the work done. Whilst the third term represents the 
expenditure of work due to the weights of the wheels of the 
train, and is wholly independent of the work done, but only 
upon the space S, through which that work is done at the 
point where the driving pressure is applied to the train. 



20 



306 FRICTION OF THE AXES OF A TRAIN. 



236. The expenditure of work due to the friction of the teeth. 

The work expended upon the friction of the teeth is repre- 
sented by the formula 

(— + — + — +... + — ) sin. 9 IT . . (282), 
x n, n, n 3 n, p ! Y ) 2 v y > 

whose value is evidently less as the factor sin. <p is less, or as 
the coefficient of friction between the common surfaces of the 
teeth is less ; and as the numbers of the teeth in the different 
wheels which compose the train are greater. The number 
of teeth in any one wheel of the train may, in fact, be taken 
so small, as to give this formula a considerable value as com- 
pared with U 2 , or to cause the expenditure of work upon the 
friction of the teeth to amount to a considerable fraction of 
the work yielded by the train : and the numbers of teeth 
of two or more wheels of such a train might even be taken 
so small as to cause the work expended upon their friction to 
equal or to surpass by any number of times the work yielded 
by the train at its working point. This will become the 
more apparent if we consider that the surfaces of contact of 
the teeth of wheels are for the most part free from unguent 
after they have remained any considerable time in action, so 
that the limiting angle of resistance assumes in most cases 
a much greater value at the surfaces of the teeth of the 
wheels than at their axes. From this consideration the 
importance of assigning the greatest possible number of 
teeth to the wheels of a train individually and collectively 
is apparent. 



237. The expenditure of work due to the friction of the axes. 
This expenditure is represented by the formula 

(U sin - * + §£ sin - '• + • • • ^gp+^fa ■ ■ ■ ** 

forming the second term of formula 280. Xow, evidently, the 
value of this formula is less as the quantities sin. <?„ sin. <p„ 
<fcc. are less, or as the limiting angles of resistance between 
the surfaces of the axes and their bearings are less, or the 




FRICTION OF THE AXES OF A TRAIN. 307 

■ubrication of the axes more perfect ; and it is less as the 

. ,. L,Pl \h L 3p3 « , 

Tactions , , , &c. are less. 

<Vy r 2 ry r A ?y 

Now, L 2 being the distance between the point of contact b 

of the third and fourth wheels 

and the projection of the point of 

contact a of the first and second 

upon the plane of those wheels, 

it follows that, generally, L 2 is 

least when the projection of a 

falls on the same side of the axis 

as the point b ;* and that it 

is least of all when this line falls on that side and in the line 

joining the axis with the point b\ whilst it is greatest of all 

when tt falls in this line produced to the opposite side of the 

axis. In the former case its value is represented by r % —r„ 

and in the latter by r 3 +?* 2 ; so that, generally, the maximum 

and minimum values of L 2 are represented by the expression 

t % ±r 2 , and the maximum and minimum values of — ^. 3 - by 
f — -£- — j p 2 . And similarly it appears that the maximum and 
minimum values of — ^i are represented by( — -{- — j p 3 ; and 

'4 '6 * f~f ' 6 ^ 

so of the rest. So that the maximum and minimum values of 
the work lost by the friction of the axes are represented 
by the expression 

\\— ± — ) Pi s^. <p x + (_ ± —\ p 2 sin. 9, + 

p 8 sin.<p 3 + lu a ; 



£±n) 



from which expression it is manifest, that in every case the 
expenditure of work due to the friction of the axes is less as 
the radii of the axes are less when compared with the radii 
of the wheels ; being wholly independent of actual dimensions 
of these radii, but only npon the ratio or proportion of the 
radius of each axis to that of its corresponding wheel : more- 

* This important condition is but a particular case of the general principle 
established in Art. 168. ; from which principle it follows, that the driving 
pressure on each wheel should be applied on the same side of the axis as the 
driven pressure. 



THE WEIGHTS OF THE WHEELS. 

"\vr, that this expenditure of work is the least when the 
wheels of the train are so arranged, that the projection of the 
point of contact of any pair upon the plane of the next 
following pair shall lie in the line of centres of this last pair, 
between their point of contact and the axis of the driving 
wheel of the pair ; whilst the expenditure is greatest when 
this projection falls in that line but on the other side of the 
axis. The difference of the expenditures of work on the 
friction of the axes under these two different arrangements 
of the train is represented by the formula 

2 | II sin. p, -f h. sin. o, + A sin. 5, + A sin. ? 4 -f . . \ XL : 
( 7\ r, 7\ r, ) 

which, in a train of a great number of wheels, may amount 
to a considerable fraction of U, ; that fraction of U, repre- 
senting the amount of power which, may be sacrificed by a 
false arraDgement of the points of contact of the wheels. 



238. The expenditure of work due to the weights of the sev-rral 
wheels of the train. 

The third and last term X . S of the expression (280) repre- 
sents the expenditure of work due to the weights of the 
several wheels of the train ; of this term the factor X is 
represented by an expression (equation 279), each of the 
terms of which involves as a factor one of the quantities N„ 
N,o N„ &c, whose general type or form is that given in 
equation (247), it being observed that the direction of the 
driving pressure on any pair of the wheels being supposed 
that of a tangent to their point of contact ; the case is that 
discussed in the note to page 266. The other factor of each 
term of the expression (equation 279) for X, is a fraction 
having the product n, n, . . . of the numbers oi teeth in all 
the preceding drivers of the train, except the first, for its 
numerator, and the product n,.n 4 .n f .., of the numbers of 
teeth in the preceding followers of the train for its denomi- 
nator : so that if the train be one by which the motion is to 
be accelerated, the numbers of teeth in the followers being 
small as compared with those in the drivers, or if the multi- 
plying power of the train be great, and if the quantities 
-S ... N a , K„ <fcc, be all positive', then is the expenditure of 
work by reason of the weights of the wheels considerable, as 



MODULUS OF A TRAIN IN WHICH THE DRIVERS ARE EQUAL. 309 

compared with the whole expenditure. Since, moreover, the 
coefficients of E" l5 N 2 , N 3 , &c, in the expression for ~N (equa- 
tion 279) increase rapidly in value, this expenditure of work 
is the greatest in respect to those wheels of the train which 
are farthest removed from its first driving wheel : for which 
reason, especially, it is advisable to diminish the weights 
of the wheels as they recede from the driving point of the 
train, which may readily be done, since the strain upon each 
successive wheel is less, as the work is transferred to it under 
a more rapid motion. 



239. The modulus of a train in which all the drivers are 
equal to one another and all the followers, and in which 
the points of contact of the drivers and followers are all 
similarly situated. 

The numbers of teeth in the drivers of the train being in 
this case supposed equal, and also the radii of these wheels, 
n 1 =n s =n b =n,=&c, r 1 —r z =r b =r,=&c. The numbers of 
teeth in the followers being also equal, and also the radii of 
the followers n q =n 4 =n 6 =&c, r^=r i =r 6 —&o,. 

If, moreover, to simplify the investigation, the driving 
work TJ, be supposed to be done upon the first wheel of the 
9 train at a point situated in re- 

spect to the point of contact a of 
that wheel with its pinion pre- 
cisely as that point of contact is 
in respect to the point of contact 
b of the next pair of wheels of 
the train ; and if a similar sup- 
position be made in respect to 
the point at which the driven work U 2 is done upon the last 
pinion of the train, then, evidently, L 1 =L 2 =L 3 = . . . =L P , 
and (see equation 247) 'N l ='N^= . . . =~N P . 

The modulus (equation 280) will become, these substitu- 
tions being made in it, the axes being, moreover, supposed 
all to be of the same dimensions and material, and equally 
lubricated, and it being observed that the drivers and the 
followers are eachp in number, 

U - jl + ^(I + Ij s ; n . ?+i? ^sm. 9l Ju i + NS . . . .(284), 
vrhich is the modulus required. 




310 



THE TKALN OF LEAST RESISTANCE. 



Moreover, the value of N (equation 279; will become by 
the like substitutions, 



»=».<2) 



© 



(285). 



The Train of least Resistance. 

240. A train of equal driving wheels and equal follower* 
being required to yield at the last wheel of the train a 
given amount of work U„ under a velocity m times areater 
or less than that under which the work U, which driven the 
train is done by the moving power upon the first wheel; it 
is required to determine what should be the number p of 
pairs of wheels in the train, so that the work U x expended 
through a given space S, in driving it, may be a min im um. 

Since the number of revolutions made by the last wheel 
of the train is required to be a given multiple or part of the 
number of revolutions made by the first wheel, which mul- 
tiple or part is represented by m, therefore (equation 231), 



m 



/n x \P n, p r x 

= ( — ) ; .*. — = m, = — ; 

\nj ' n, r, 



••• £+=) 



m 



m 
l 

""+1 



, 1 m 

and — = — 



», 



and — = — r 



r,r. 



Substituting these values in the modulus (equation 284); 
substituting, moreover, for K its value from equation (285), 
we have 



1 /T I 

U,= \l+—p(m p -f-l)sin.<p+ I— x ~i\ pm p sin. <p,j 



THE TRAIN OF LEAST RESISTANCE. 311 



U.+NXm-l)! -^ )S (286\ 






It is evident that the question is solved by that value of p 
which renders this function a minimum, or which satisfies 

the conditions -— L = and -^-r > 0. The first condition 
dp dp 2 If- 

gives by the differentiation of equation (286), 

j-" P (l- l£ ^)g-n. 9 + ^sin. ? ,) + Jsin. 9 | 

1 

U a+ m " y(m - 1)1 °^^ 1 ^0....(287). 

p\ m *-iy 

This equation may be solved in respect to p, for any given 
values of the other quantities which enter into it, hy approxi- 
mation. If, being differentiated a second time, the above 
expression represents a positive quantity when the value of 
p (before determined) is substituted in it, then does that 
value satisfy both the conditions of a minimum, and sup- 
plies, therefore, its solution to the problem. 

If we suppose ^=0 and 1^=0, or, in other words, if we 
neglect the influence of the friction of the axes and of the 
weights of the wheels of the train upon the conditions of the 
question, we shall obtain 

m W i _ ^-1— _ 6 i n . <p + _ S i n . p =0 ; 
\ p ]n x n, T 

whence by reduction, 

p^ 1 ^* (288). 



1 + m 



p 



* This formula was given by the late Mr. Davis Gilbert, in his paper on the 
"Progressive improvements made in the efficiency of steam engines in Corn- 
wall," published in the Transactions of the Royal Society for ISM. Towards 
the conclusion of that paper, Mr. Gilbert h;is treated of the methods best 
adapted for imparting great angular velocities, and, in eonneetion with that 
subject, of the friction of toothed wheels; Laving reference to the friction of 
the surfaces of their teeth alone, and neglecting all consideration of the inllu- 



312 THE INCLINED PLANE. 



The Inclined Plane. 

241. Let AB represent the surface of an inclined plane on 
which is supported a body whose centre of gravity is C, and 
its weight W, by means of a pressure acting in any direction, 
and which may be supposed to be supplied by the tension of 
a cord passing over a pulley and carrying at its extremity 
a weight. 

Let OR represent the direction of the resultant of P and 
W. If the direction of this line be inclined to the perpen- 
dicular ST to the surface of the plane, at an an^le OST 
equal to the limiting angle of resistance, on that sicfe of ST 
which is farthest from the summit B of the plane (as in 
fig. 1), the body will be upon the point of slipping upwards; 
and if it be inclined to the perpendicular at an angle OST, 

ence due to the weights of the wheels and to the friction of their axes. The 
author has in vain endeavoured to follow out the condensed reasoning by which 
Mr. Gilbert has arrived at this remarkable result ; it supplies another example 
of that rare sagacity which he was accustomed to bring to the discussion of 
questions of practical science. Mr. Gilbert has given the following examples 
of the solution of the formula by the method of approximation:— If ?/i=120, 
or if the velocity is to be increased by the train 120 times, then the value of/) 
given by the above formula, or the number of pairs of wheels which should 
compose the train, so that it may work with a minimum resistance, reference 
being had only to the friction of the surfaces of the teeth, is 3 - 745 ; and the value 

< f the factor p(m p -\-\) (equation 286), which being multiplied bv — sin. <f> U 2 

r ^presents the work expended on the friction of the surfaces of the teeth, is in 
this case 17*9 ; whereas its value would, according to Mr. Gilbert, be 121 if the 
velocity were got up by a single pair of wheels. So that the work lost by the 
friction of the teeth in the one case would only be one seventh part of that in 
the other. In like manner Mr. Gilbert found, that if ?nz=100, then p should 
equal 3*6 ; in which case the loss by friction of the teeth would amount to the 
sixth part only of the loss that would result from that cause if p=l, or if the 
required velocity were got up by one pair of wheels. 

If m=40, then ^=2-88, with a gain of one third over a single pair. 

If m=3-59, thenp=l. 

Ifm=12-85, thenj»=2. 

Ifw=46-3, thenp=3. 

Ifwi=166-4, then/>=4. 

It is evident that when p, in any of the above examples, appears under the 
form of a fraction, the nearest whole number to it, must be taken in practice. 
The influence of the weights of the wheels of the train, and that of the friction 
of the axes, so greatly however modify these results, that although they are 
fully sufficient to show the existence in every case of a certain number of 
wheels, which being assigned to a train destined to produce a given accelera- 
tion of motion shall cause that train to produce the required effect with the 
least expenditure of power, yet they do not in any case determine correctlj 
what that number of wheels should be. 



THE INCLINED PLANE. 



313 




(20 




equal to the limiting angle of resistance, but on the side of 
ST nearest to the summit B (as in fig. 2.), then the body will 
be upon the point of slipping downwards (Art. .138.); the 
former condition corresponds to the superior and the latter 
to the inferior state bordering upon motion (Art. 140.). 

!Now the resistance of the plane is equal and opposite to 
the resultant of P and W ; let it be represented by P. 

There are then three pressures P, W, and P in equili- 
brium. 



.-. (Art. 14.) 



sm. 



WOP 



W sm. 



POP. 



Let /BAC=«, ZOST=lime. / f resistance— <p, let d 
represent the inclination PQB of the direction of P to the 
surface of the plane, and draw OY perpendicular to AB ; 
then, 



in fig. 1, WOK=WOy + SOV=BAC + OST=i + <p, 

and POB=PQB + OSQ=PQB+^-OST=*-M-p ; 

2 2i 

in fig. 2., WOP=WOY-SOY=BAC-OST=»-9, 
and POR=PQB + OSQ: 

.\WOR=» + 9; 



:PQB + *+OST=*+d+9 



and 



POP=- + 



+*)> 



the upper or lower sign being taken according as the body 
is upon the point of sliding up the plane, as in fig. 1, or 
down the plane, as in fig. 2. Or if we suppose the angle <p 
to be taken positively or negatively according as the body is 
on the point of slipping upwards or downwards ; then gene- 



rally W R =< -f- <p 



POR=- + < 



'-9); 



314 



THE INCLINED PLANE. 



P 

w 



sili. (< + 9) _sin. (« + <p) . 
cos. (0— 9 ' 



in -(i+*-*) 



' cos.(d— 9) 



(289). 



If the direction of P be parallel to the plane, / PQB 01 
0=0 ; and the above relation becomes 

y=w . Bin -(' + <p ) 

COS. <p 
(8.) 





If »=0 the plane becomes horizontal (fig. 3)., and the re- 
lation between P and TV assumes the form 



P=W. 



sin. <p 



(291). 



cos. (^—9) 

If (9=0, P=TV . tan. <p, as it ought (see Art. 138.). 

If the angle PQB or 6 (fig. 1.) be increased so as to be- 
come tf— 0, PQ will assume the direction shown in fig. 4, 
and the relation (equation 289), between P and TV will be- 
come 



p = _tv 6in - 1 ,+9 ) 

cos. (6 + 9) 



(292). 



The negative sign showing that the direction of P must, 
in order that the body may slip up the plane, be opposite to 
that assumed in fig. 1. ; or that it must be a pushing pres- 
sure in the direction PO instead of a pulling pressure in the 
direction OP. 

If, however, the body be upon the point of slipping down 
the plane, so that 9 must be taken negatively ; and if, more- 
over, 9 be greater than », then sin. (» + <p), will become sin. 
(<— 9)— — sin. (9—')) bo that P will in this case assume the 
positive value 



P =w . - g in -£-'| 

cos. {?—<?) 



(293), 



THE MOVEABLE INCLINED PLANE. 



315 



which determines the force just necessary under these cir 
cumstances to pull the body down the plane. 

If j=<p, P=0, the body will therefore, in this case, be upon 
the point of slipping down the plane without the application 
of any pressure whatever to cause it to do so, other than its 
own weight. The plane is under these circumstances, said 
to be inclined at the angle of repose, which angle is there- 
fore equal to the limiting angle of resistance. 



242. The direeUon of least traction. 

Of the infinite number of different directions in which the 
pressure P may be applied, each requiring a different amount 
to be given to that pressure, so as to cause the body to slide 
up the plane, that direction will require the least value to be 
assigned to P for this purpose, or will be the direction of 
least traction, which gives to the denominator of the fraction 
in equation (289) its greatest value, or 
which makes d— <p=0 or d=<p. The di- 
rection of P is therefore that of least 
traction when the angle PQB is equal to 
the limiting angle, a relation which ob- 
tains in respect to each of the cases dis- 
jl cussed in the preceding article. 




243. The Moveable Inclined Plane. 



Let ABC represent an inclined 
plane, to the back AC of which 
is applied a given pressure P„ 
and which is moveable between 
the two resisting surfaces GH and 
KL, of which either remains fixed, 
and the other is upon the point 
of yielding to the pressure of the 
plane. 

If we suppose the resultants of the resistances upon the 
different points of the two surfaces AB and BC of the plane 
to be represented by R, and R a respectively, it is evident 
that the directions of these resistances and of the pressure I\ 




316 THE MOVEABLE INCLINED PLANE. 

will meet, when produced, in the same point O* ; and that, 
since the plane is upon the point of slipping upon each of 
the surfaces, the direction of each of these resistances is 
inclined to the perpendicular to the surface of the plane, at 
the point where it intersects it, at an angle equal to the cor- 
responding limiting angle of resistance. 

80 that if ET and FS represent perpendiculars to the 
surfaces AB and BC of the plane at the points E and F and 
<p„ <p 2 , the limiting angles of resistance between these surfaces 
of the plane and the resisting surfaces GH and KL re- 
spectively, then RjETmpj, R 2 FS=? 2 . 

Now the pressures P„ Ii„ R 2 being in equilibrium (Art. 

P, s in. EOF P 1 _sm 1 EOF 

K.-sin.DOF' and K 2 ~sin.DOE' 

But the four angles of the quadrilateral figure BEOF 
being equal to four right angles (Euc. 1*32), EOF=2*r— 

EBF-OEB-OFB; but EBF=«, OEB^-fP, OFB= 
- + (p 2 . .\EOF=*'— 1— <p i — <p a . 

Similarly, DOE=2rr-ADO-AEO-DAE ; but ADO= 
J AEO=J- 9l , BAC=^-.: .-.DOE^+i+fr 



Since, moreover, DO is parallel to BC, both being per- 

2 



pendicular to AC, .\DOF=*-OFC ; but OFC=^-<p, : 



.-.DOF=g+9,. 



P 1 _ sin. fo— (' + <?,+<)} sin. (i + ^ + yQ 
R~ . /t , m \ cos. <p 9 



sin 



•(§+'•) 



, p=ES in.(. + 9, + 9,) ( 

1 ' cos. <p 2 

P t sin. fo— Q + ^ + Qi _ sin. + ^ + 9, ). 

R,~~ . I* , , ^ \ cos. + 9,) ' 

sin.l- + ' + ? 1 ] 

Since either is equal and opposite to the resultant of the other two. 



A SYSTEM OF TWO MOVEABLE INCLINED PLANES., 317 

smj^+j^ 

1 cos. (»+9j 

In the case in which the surface GrH yields to the pressure 
of tl;e plane, KL remaining fixed, we obtain (equation 121.) 
for the modulus (see Art. 148.) observing that P 1 (°)=E. X sin. » 
(equation 294), 

sin. (, + „, + ?,) 

sin. i . cos. 9 2 

In the case in which the surface KL yields, CH remaining 
fixed, observing that P 1 (°)=E 3 tan. i (equation 295), we have 

sin -0 +?,+»■> (297). 

cos.(« + 9i)tan.» 

Equations (296) and (297) may be placed respectively un- 
der the forms 

and U =U cos '.( <p > +<p «\l t an.i + tan.fa + tp.) ) 

1 a sin. <p x ((cot. cp x — tan. i) tan. i j ' 

The value of JJ l corresponding to a given value of U a is in 

ft 
the former equation a minimum when «=-, and in the latter 

when 

tan. i= \ a/-. C °? ,(P ; , . -1 1 tan. (?,+?,) (298). 

[y sin. ^sm.^ + fp,) ) v * Tsy v y 

From the former of these equations it follows, that the work 
lost by friction (when the driving surface of the plane is its 
hypothenuse) is less as the inclination of the plane is greater, 
or as its mechanical advantage is less. 



244. A system of two moveable inclined planes. 
Let A and B represent two inclined planes, of which A 



318 A SYSTEM OF TWO MOVEABLE INCLrNED PLANES. 

p rests upon a horizontal surface, and 

J * receives a horizontal! motion from 

B | ; 1>""*r * ne ac *i on °f tne pressure P, ; com- 

v^..':::::.'3r/......„1...8 * municating to B a motion which is 

A "p 1 I^^__ restricted to a vertical direction by 

^ ^•-•^ p 1^ the resistance of the obstacle D, 

\ \ ( which vertical motion of the plane 

^ J~ I=r '~ is opposed by the pressure P 2 ap- 

* plied to its superior surface. It is 

required to determine a relation between the pressures T t 
and P„ in their state bordering upon motion ; and the mo- 
dulus of the machine. 

Let E x represent the pressure of the plane A upon the 
plane B, or the resistance of the latter plane upon the former, 
and E 3 the resistance of the obstacle D upon the back of the 
plane B ; then is the relation between Ej and P x determined 
by equation (294). And since E„ E 3 , P 2 are pressures in 
equilibrium, the relation between E, and P, is expressed 

(Art. 14.) by the relation?- 1 = ^-^5- 3 . Kow E 3 Q is 
1 J P 2 sin. P 1 QP 3 

inclined to a perpendicular to the back of the plane B, at an 
angle equal to the limiting angle of resistance between the 
surface of that plane and the obstacle D on which it is upon 
the point of sliding. Let this angle be represented by <p„ 
then is the inclination of B 3 to the back of the plane or P 2 Q 

represented by-— p 3 ; so that P 2 QE 3 =-— <p,. 

And if E 3 Q be produced so as to meet the surface of the 
plane A in V, and VS be drawn horizontally, PjQR^ 

QVR I +VE 1 Q=E 1 VSH-SYA+VR I Q=<P, + '+g+9 I , 

where » represents the inclination of the superior surface of 
the plane A or the inferior surface of the plane B to the 
horizon. Substituting these values of P a QE 3 and E,QE, wo 
obtain 

E t Sin '(j-^) _ cos. 9, 

sin.| +i + <p, + 9,j 

Multiplying this equation by equation (294), and solving in 
respect to P„ 



A SYSTEM OF THREE INCLLYED PLANES. 



319 



p _p 8m - (* + ?! + <?„) cos. <p, 
2 COS. (« + <?! + 9s) cos - 9 2 



(299). 



•.(Art. 152.) Tj-TJ^ sin.fr+9, + 9,) cos. 9 3 ^ # ^ # (3(M)) 
cos. (*4-9x + 9 3 ) tan. t cos.9 a 




A system of three inclined planes, two of which are movea- 
ble, and the third fixed. 

245. The inclined plane A, in the accompanying figure, is 
fixed in position, the plane B is 
moveable upon A, having its upper 
surface inclined to the horizon at a 
less angle than the lower ; and C is 
an inclined plane resting upon B, 
which is prevented from moving 
horizontally by the obstacle D, but 
may be made to slide along this 
obstacle vertically. It is required 
to determine a relation between 
P, and P 2 , applied, as shown in the figure, when the system 
is in the state bordering upon motion. 

Let R„ B 2 , R 3 represent the resistances of the surfaces on 
which motion takes place, <j>, 9 2 9 3 their limiting angles of 
resistance respectively, and i 1? * 2 the inclinations of the two 
surfaces of contact of B to the horizon. Since P 1? R 1? R 2 are 
pressures in equilibrium, as also P 2 , R 2 , B 3 

. P, sin. B 2 OR 1 R 2 _ sin. P 2 QR 3 
" R 2 ~ sin. P x OK x ' P 3 ~ sin. R 2 Q£V 

Multiplying these equations together, 

P, sin. R.OB, . sin. P 2 QR, 
P, - sin. P.OR, . sin. R 2 QR 3 

Draw OS and OT parallel to the faces of the plane B ; then 

R.OR^^OS + QOT-TOS; but R.OS^ _ -<p x , since OS ie 

if 
parallel to the inferior face of the plane B, also QOT= — q> a , 

since OT is parallel to the superior face of the plane B; and 



320 A SYSTEM OF THREE INCLINED PLANES. 

TOS = the inclination of the faces of the plane B to one 
another =ij — i t . 

Also P 2 QR 3 = I -P,QM= I - <p,. 

LetP.O be produced to Y; therefore P 1 OR l =^—E 1 OY= 

<r-(R 1 OS-SOV)=«r- j g -9.) -«, [ = | + ', + 9, Lastly 
R 2 QK 3 = OQM + MQK 3 . Now, MQR 3 =9 3 ; also, OQM = 
r«QOV=*-(QOT+TOV)=r-jg- Vl )+i i l=~. i + 9iS 



/.R a QK 3 =--« 2 +9 s +9,= £-(' 3 -9 2 -9 3 )- 



Bin 
. ^u 

sm. 



(g+VHfc) • sin - j g -0,-^-^) j 



p _p 2 sm. j(^ + cp ) + (, ,)} cos. ^ 

2 cos. («, + ?,) cos. {»,— (9 a + 9,)f } 

Wlience we obtain for the modulus (Art. 152.\ observing 

tb a t* 1 w= sin - (, -'- ) . 

cos. », cos. I, 

u =IJ sin. fa + 9 3 + *,—*,) cos- '1 co s. i % cos.9 8 (302 ^ 

2 cos.(t a — 9 a — 9 s )cos.(* 1 +9 1 )sin.(' 1 — < 8 ) * 



THE WEDGE DRIVEN BY PRESSURE. 



321 



The Wedge driven by Pressure. 

246. Let ACB represent an isosceles wedge, whose angle 
ACB is represented by 2i, and which is 
driven between the two resisting surfaces 
DE and DF, by the pressure P a . Let R, 
and R 2 represent the resistances of these 
surfaces upon the acting surfaces CA and 
CB of the wedge when it is upon the 
point of moving forwards. Then are the 
directions of R, and R 2 inclined respec- 
tively to the perpendicular Gs and R£ 
to the faces CA and CB of the wedge, at 
angles each equal to the limiting angle of 
resistance 9. The pressures B^ and R 3 are 
therefore equally inclined to the axis 
of the wedge, and to the direction of P„ whence it follows 
that R 1 =R a , and therefore (Art. 13.) that P 1 =2R 1 cos. JGOR. 
Now, since CGOR is a quadrilateral figure, its four angles 
are equal to four right angles ; therefore GOR=2tf— GCR— 




OGC-ORC. ButGCR=2*; OGC=ORC 



2+* : 



.-. GOR=*--(2t-r-2<p) .-. £GOR: 



(<+9). 



wedge is 



,\P 1 =2R 1 sin.(<+<p) (303). 

Whence it follows (equation 121) that the modulus of the 

n,=u, sin -. (t+y) (304). 

sin. 1 
This equation may be placed under the form 

U^U,, {cot. 9 + cot. i\ sin. 9. 

The work lost by reason of the friction of the wedge is 
greater, therefore, as the angle of the wedge is less ; and 
infinite for a finite value of 9, and an infinitely small value 
of 1. 



The angle of tJie wedge. 

247. Let the pressure P 15 instead of being that just sufli- 
21 



322 



THE WEDGE DRIVEN BY PRESSURE. 



cient to drive the wedge, be now supposed 
to be that which is only just sufficient tc 
keep it in its place when driven. The two 
surfaces of the wedge being, under these 
circumstances, upon the point ot sliding 
backwards upon those between which the 
wedge is driven, at their points of contact 
G and E, it is evident that the directions 
of the resistances ifi and i a R upon those 
points, must be inclined to the normals 
6-G and t~R at angles, each equal to the 
limiting angle of resistance, but measured 
on the sides of those normals opposite to 

those on which the resistances R,G and RJl are applied.* 
In order to adapt equation (303) to this case, we have 

only then to give to 9 a negative value in that equation. It 

will then become 




P.=2R 



sin. 



(<-<?) 



(305). 



So long as 1 is greater than <p, or the angle C of the wedge 
greater than twice the limiting angle of resistance, P, is 
positive ; whence it follows that a certain pressure acting in 
the direction in which the wedge is driven, and represented 
in amount by the above formula, is, in this case, necessary 
to keep the wedge from receding from any position into 
which it has been driven. So that if, in this case, the pres- 
sure P, be wholly removed, or if its value become less than 
that represented by the above formula, then the wedge will 
recede from any position into which it has been driven, or 
it will be started. If 1 be less than <p, or the angle C of the 
wedge less than twice the limiting angle of resistance, Y x 
will become negative ; so that, in this case, a pressure, oppo- 
site in direction to that by which the wedge has been driven, 
will have become necessary to cause it to recede from the 
position into which it has been driven ; whence it follows, 
that if the pressure P, be now wholly removed, the wedge 
will remain fixed in that position ; and, moreover, that it 
will still remain fixed, although a certain pressure be applied 
to cause it to recede, provided that pressure do not exceed 
the negative value of P l5 determined by the formula. 



* This will at once be apparent, if we consider that the direction of the 
resultant pressure upon the wedge at G must, in the one case, be such, that if 
it acted alone, it would cause the surface of the wedge to slip downwards on 
the surface of the mass at that point, and in the other case upwards; and that 
the resistance of the mass is in each case opposite to this resultant pressure. 



THE WEDGE DRIVEN BY IMPACT. 323 

It is this property of remaining fixed in any position into 
which it is driven when the force which drives it is removed, 
that characterises the wedge, and renders it superior to 
every other implement driven by impact. 

It is evidently, therefore, a principle in the formation of a 
wedge to be thus used, that its angle should be less than 
twice the limiting angle of resistance between the material 
which forms its surface, and that of the mass into which it 
is to be driven. 



The Wedge driven by Impact. 

248. The wedge is usually driven by the impinging of a 
heavy body with a greater or less velocity upon its back, in 
the direction of its axis. Let W represent the weight of 
such a body, and Y its velocity, every element of it being 
conceived to move with the same velocity. The work 
accumulated in this body, when it strikes the wedge, will 

1 "W 

then be represented (Art. 66.) by - — Y 2 . Now the whole of 

this work is done by it upon the wedge, and by the wedge 
upon the resistances of the surfaces opposed to its motion ; 
if the bodies are supposed to come to rest after the impact, 
and if the influence of the elasticity and mutual compression 
of the surfaces of the striking body and of the wedge are 
neglected, and if no permanent compression of their surfaces 

1 WY 2 

follows the impact.* .*. TJ l =- . 



* The influence of these elements on the result may be deduced from the 
principles about to be laid down in the chapter upon impact. It results from 
these, that if the surfaces of the impinging body and the back of the wedge, 
by which the impact is given and received, be exceedingly hard, as compared 
with the surfaces between which the wedge is driven, then the mutual pressure 
of the impinging surfaces will be exceedingly great as compared with the 
resistance opposed to the motion of the wedge. Now, this latter being 
neglected, as compared with the former, the work received or gained by the 
wedge from the impact of the hammer will be shown in the chapter upon 

impact to be represented bv > ' — , where Wi represents the 

weight of the hammer, W 2 the weight of the wedge, and e that measure of 
the elasticity whose value is unity when the elasticity is perfect. Equating 
this expression with the value of Ui (equation 304), and neglecting the effect 
of the elasticity and compression of the surfaces G and R, between which the 
wedge is driven, we shall obtain the approximation 

(l-J-eywyWaV sin, i 
a ~ 20(W,+W 9 ) 9 sin. («+*)* 



324 



THE WEDGE DRIVEN BY IMPACT. 



Substituting this value of U, in equation 304, and solving in 
respect to U„ we have 



CT,= 



1 WY* sin. i 



2 g sin. (1+9) 



(306); 



by which equation the work TJ a yielded upon the resistances 
opposed to the motion of the wedge by the impact of a given 
weight W with a given velocity V is determined ; or the 
weight "W necessary to yield a given amount of work when 
moving with a given velocity ; or, lastly, the velocity V with 
which a body of given weight must impinge to yield a given 
amount of work. 

If the wedge, instead of being isosceles, be of the form of 
a right angled triangle, as shown 
in the accompanying figure, the 
relation between the work U , done 
upon its back, and that yielded 
upon the resistances opposed to 
its motion at either of its faces, is 
represented by equations (296) 
and (297). Supposing therefore 
this wedge, like the former, to be 
driven by impact, substituting as before for JJ 1 its value 

1 "W 

V 2 , and solving in respect to U a , we have in the case in 

if 

which the face AB of the wedge is its driving surface 




U.= 



1 TTY a sin. i cos. <p, 



(307); 



*~2 g ' sin. (* + <?! + <p,) 

when the base BC of the wedge is its driving surface, 

jj 1WV tan. * cos. (* + $,) /gQgN 

3_ 2 g ' Bin.(t+9,+90 



From this expression it follows, that the useful work is the greatest, othei 
things being the same, when the weight of the wedge is equal to the weight 
of the hammer, and when the striking surfaces are hard metals, so that the 
Talue of e may approach the nearest possible to unity. 



THE MEAN PRESSURE OF IMPACT. 



325 




respect to U a , we have 

cos, (s-fg^+tp^tan. » cos. <p a 
g sin. (i + 9 x +9 2 ) cos <p, 



249. If the power of the wedge 
be applied by the intervention of 
an inclined plane moveable in a 
direction at right angles to the di- 
rection of the impact*, as shown in 
the accompanying figure, then sub- 
stituting for U, in equation (300) 
half the vis viva of the impinging 
body, and solving, as before, in 



1W 



(309). 




IT _1W cos^ 
a ~2 g 



If instead of the base of the 
__jm plane being parallel to the direc- 
tion of impact, it be inclined to 
it, as shown in the accompanying 
figure, then, substituting as above 
in equation 302, we have 

-9 2 -9 3 ) cos. fc + qQ sin. fo—Q ^ ^ ^^ 
sin. (9 1 + 9, + * 1 — cos. c 1 cos. * 2 cos.<p 3 ' ' ' ' ^ '" 



The mean Pressure of Impact. 

250. It is evident from equations 306, 307, 308, that, since, 
whatever may be the weight of the impinging body or the 
velocity of the impact, a certain finite amount of work U 3 is 
yielded upon the resistances opposed to the motion of the 
wedge ; there is in every such case a certain mean resistance 
E overcome through a certain space S, in the direction in 
which that resistance acts, which resistance and space are 
such, that 

KS=U 2 , and therefore R=-o-*. 

b 

If therefore the space S be exceedingly small as compared 



* This is the form under which the power of the wedge is applied for the 
expressing of oil. 



326 the screw. 

with U„ there will be an exceedingly great resistance E 
overcome by the impact through that small space, however 
slight the impact. From this fact the enormous amount of 
the resistances which the wedge, when struck by the ham- 
mer, is made to overcome, is accounted for. The power of 
thus subduing enormous resistances by impact is not how- 
ever peculiar to the wedge, it is common to all implements 
of impact, and belongs to its nature ; its effects are rendered 
« r/nanent in the wedge by the property possessed by that 
implement of retaining permanently any position into which 
it is driven between two resisting surfaces, and thereby op- 
posing itself effectually to the tendency of those surfaces, by 
reason of their elasticity, to recover their original form and 
position. It is equally true of any the slightest direct impact 
of the hammer as of its impact applied through the wedge, 
that it is sufficient to cause any finite resistance opposed to 
it to yield through a certain finite space, however great that 
resistance may be. The difference lies in this, that the sur- 
face yielding through this exceedingly small but finite space 
under the blow of the hammer, immediately recovers itself 
after the blow if the limits of elasticity be not passed ; 
whereas the space which the wedge is, by such an impact, 
made to traverse, in the direction of its length, becomes a 
permanent separation. 



The Screw. 

251. Let the system of two moveable inclined planes re- 
presented in fig. p. 318. be formed of ex- 
ceedingly thin and pliable laminae, and con- 
ceive one of them, A for instance, to be 
wound upon a convex cylindrical surface, as 
shown in the accompanying figure, and the 
other, B, upon a concave cylindrical surface 
having an equal diameter, and the same axis 
with the other ; then will the surfaces 
EF and GH of these planes represent truly 
the threads or helices of two screws, one of them of the form 
called the male screw, and the other the female screw. Let 
the helix EF be continued, so as to form more than one spire 
or convolution of the thread ; if, then, the cylinder which 
carries this helix be made to revolve upon its axis by the 
action of a pressure P, applied to its circumference, and the 
cylinder which carries the helix GH be prevented from re- 




THE SCREW. 327 

volving upon its axis by the opposition of an obstacle D, 
which leaves that cylinder nevertheless free to move in a 
direction parallel to its axis, it is evident that the helix EF 
will be made to slide beneath GH, and the cylinder which 
carries the latter helix to traverse longitudinally ; moreover, 
that the conditions of this mutual action of the helical sur- 
faces EF and GH will be precisely analogous to those of the 
surfaces of contact of the two moveable inclined planes dis- 
cussed in Art. 244. So that the conditions of the equili- 
brium of the pressures P x and P 2 in the state bordering upon 
motion, and the modulus of the system, will be the same in 
the one case as in the other ; with this single exception, that 
the resistance R 2 of the mass on which the plane A rests (see 
fig. p. 318.) is not, in the case of the screw, applied only to 
the thin edge of the base of the lamina A, but to the whole 
extremity of the solid cylinder on which it is fixed, or to a 
circular projection from that extremity serving it as a pivot. 
Now if, in equation 299, we assume <p 2 =0, we shall obtain 
that relation of the pressures P, and P 2 in their state border- 
ing upon motion, which would obtain if there were no fric- 
tion of the extremity of the cylinder on the mass on which it 
rests ; and observing that the pressure P 2 is precisely that 
by which the pivot at the extremity of the cylinder is pressed 
upon this mass, and therefore the moment (see Art. 176, 
equation 183) of the resistance to the rotation of the cylinder 

2 
produced by the friction of this pivot by -P 2 p tan. <p 2 , where 

o 

p represents the radius of the pivot ; observing, moreover, 
that the pressure which must be applied at the circumfe- 
rence of the cylinder to overcome this resistance, above that 
which would be required to give motion to the screw if there 

were no such friction, is represented by- P 2 -tan. <p 9 , r being 

taken to represent the radius of the cylinder, we obtain for 
the entire value of the pressure P x in the state bordering 
upon motion 

p sin.(* + <p 1 )cos.tp 2 p p 

The pressure P, has here been supposed to be applied to 
turn the screw at itewrewmfermce f it is customary, however, 
to apply it at some distance from its circumference by the 
intervention of an arm. If a represent the length of such an 



THE SCRKW. 

arm, measuring from the B - the cylinder, it is evident 
that the pi ae P, applied to the extremity of that arm, 
would produce at the circumference of the cylinder a pre* 

represented by I sh expression being substitute 

Pj in the preceding equation, and that equation solved in 
■t to P.. we obtain finally for the relation between P. 
and P, in tl. "'ordering upon motion, 

p.=p,, -e~ - •(»"> 

if in hke manner we assume in the modulus 'equation 300) 
j. = ". and thus determine a relation between the work done 
at the driving point and that yielded at the working point, 
on the supposition that no work is expended on the friction 
of the pivot : and if to the value : I" thus obtaint 
the work expended upon the resistance of the pivot which is 
shown (equation 1 - 1 -ented at each revolution 

by -~pP, tan. 5 S . and therefore during n revolutions by 
o 

4 

e shall obtain the following general expression for 
o 

the modulus : the whole expenditure of work due to the 
prejudicial resistances being taken in! nt. 

U,=L, . — ! -1 J . tan. <p r 

' c s. .---- -an. t '3 

Representing by >• the common distance between the threads 
of the screw, i. t . file space which the nut B is made to 

traverse at each revolution of the screw ; and o: 

± ± r 

ing that ;i>-P 2 = r s . so that ^'TipP, tan. ;.=-*•— *p tan. <p,= 

2 ■'-/• p . '2~r 

. -.U, tan. :,. in which expression -r— = co£ <. we 

obtain finally for the modulus of the screw 

„ ~ \ sin. \'t — - - | . 

L ,=U, • : — - - *■ (313). 

It is evidently immaterial to the result at what distance 
from the axis the obstacle D ifi Bed to the revolution of 



APPLICATIONS OF THE SCREW. 



329 






that cylinder which carries the lamina B ; since the amount 
of that resistance does not enter into the result as expressed 
in the above formula, but only its direction determined by 
the angle <p a , which angle depends upon the nature of the 
resisting surfaces, and not upon the position of the resisting 
point. 



Applications of the Scbew. 

252. The accompanying figure represents an application 
of the screw under the circumstances described in the pre- 
ceding article, to the well known machine called the Yice. 




AB is a solid cylinder carrying on its surface the thread of a 
male screw, and within the piece CD is a hollow cylindrical 
surface, carrying the corresponding thread of a female 
screw; this female screw is prevented from revolving with 
the male screw by a groove in the piece CD, which carries 
it, and which is received into a corresponding projection EF 
of the solid frame of the machine, serving it as a guide ; 
which guide nevertheless allows a longitudinal motion to 
the piece CD. A projection from the frame of the instru- 
ment at B, met by a pivot at the extremity of the male 
screw, opposes itself to the tendency of that screw to tra- 
verse in the direction of its length. The pressure P„'to be 
overcome is applied between the jaws H and K of the vice, 
and the driving pressure P x to an arm which carries round 
with it the screw AB. 

It is evident that, in the state bordering upon motion, the 
resistance B upon the pivot at the extremity B of \hv screw 
AB, resolved in a direction parallel to the length of that 
screw, must be equal to the pressure P„ (sec Art. !*>.); so 
that if we imagine the piece CD to become lixed, and the 



330 



APPLICATIONS OF THE SCREW. 




r« A .[ to become moveable, being prevented from revolv 
tog, as CD was, by the intervention of a groove and guide- 
Then might the instrument be applied I me any given 
ince R opposed to the motion of this piece CD by the 
constant pressure of its pivot upon that piece. 
The screw is applied under these circumstances in the 

common screw press. The piece 
A, fixed to the solid frame of the 
machine, contains a female screw 
whose thread corresponds to that 
of the male screw ; this screw, 
when made to turn by means of a 
handle fixed across it, presses by 
the intervention of a pivot B. at its 
extremity, upon the surface of a 
solid piece EF moveable verti- 
cally, but prevented from turning 
with the screw by grooves receiv- 
ing two vertical pieces, which 
serve it as guides, and form parts 
of the frame of the machine. 
a The formula? determined in 

Art. 251. for the preceding cast 

of the application of the screw, obtain also in this case, it 
we assume p s =0. The loss of power due to the friction of 
the piece EF upon its guides will, however, in this calcu- 
lation, be neglected ; that expenditure is in all cases exceed- 
ingly small, the pressure upon the guides, whence their 
friction results, being itself but the result of the friction of 
the pivot B upon its bearings : and the former friction being 
therefore, in all cases, a quantity of two dimensions in 
respect to the coefficient of friction. 

If, instead of the lamina A (p. 326.) being fixed upon the 
convex surface of a solid cylinder, and B upon the concave 
surface of a hollow cylinder, the order be reversed. A being 
fixed upon the hollow and B on the solid cylinder, it is evi- 
dent that the conditions of the equilibrium will remain the 
same, the male instead of the female screw being in this case 
made to progress in the direction of its length. If. however, 
the longitudinal motion of the male screw B _ be, 

under these circumstances, arrested, and that screw thus 
become fixed, whilst the obstacle opposed to the longitudinal 
motion of the female screw A is removed, and that screw 
thus becomes free to revolve upon the male screw, and also 
to traverse it longitudinally, except in as far as the latter 



THE DIFFERENTIAL SCREW. 



331 




motion is opposed by a certain resistance 
R, which the screw is intended, under 
these circumstances, to overcome ; then 
will the combination assume the well 
known form of the screw and nut. 

To adapt the formulae of Art. 251. to 
this case, <p 3 must be made = 0, and 
instead of assuming the friction upon the extremity of the 
screw (equation 311) to be that of a solid pivot, we must 
consider it as that of a hollow pivot, applying to it (by 
exactly the same process as in Art. (251.), the formulae of 
Art. (1T7.) instead of Art. (176.). 



The Differential Screw. 

253. In the combination of three inclined planes discussed 
in Art. 245., let the plane B be conceived of much greater 
width than is given to it in the figure (p. 319.), and let it 
then be conceived to be wrapped upon a convex cylindrical 
surface. Its two edges ab and cd will thus become the 
helices of two screws, having their threads of different incli- 
nations wound round different portions of the same cylinder, 





?» 


7*S 


^ 












k 


J 


tv* 8 


5S5S&E? 


-;.--/::.; 


m 


^:^fm^m 


-mmm 


r r r r i-ts l 


mote 


n 




















% 






I 






_ 













as represented in the accompanying figure, where the thread 
of one screw is seen winding upon the surface of a solid 
cylinder from A to C, and the thread of another, having a 
different inclination, from D to B. 

Let, moreover, the planes A and C (p. 319.) be imagined 
to be wrapped round two hollow cylindrical surfaces, of 
equal diameters with the above-mentioned solid cylinder, 
and contained within the solid pieces E and F, through 
which hollow cylinders AB passes. Two female screws will 
thus be generated within the pieces E and F, the helix of 



332 

the one adapting itself to that of the male screw extending 
{V«>i 1 1 A to C, and the helix of the other to that upon the 
male Bcrew extending from D to B. If, then, the piece E 
be conceived to be fixed, and the piece F moveable in the 
direction of the length of the screw, but prevented from 
turning with it by the intervention of a guide, and if a pres- 
sure P, be applied at A to turn the screw AB, the action of 
this combination will be precisely analogous to that of the 
system of inclined planes discussed in Art. 245., and the 
conditions of the equilibrium precisely the same ; so that the 
relation between the pressure P, applied to turn the screw 
(when estimated at the circumference of the thread) and that 
P 2 , which it may be made to overcome, are determined by 
equation (301), and its modulus by equation (302). 

The invention of the differential screw has been claimed 
by M. Prony, and by Mr. White of Manchester. A com- 
paratively small pressure may be made by means of it to 
yield a pressure enormously greater in magnitude.* It 
admits of numerous applications, and, among the rest, of 
that suggested in the preceding engraving. 



Hunter's Screw. 

254. If we conceive the plane B (p. 319.) to be divided 
by a horizontal line, and the upper part 
to be wrapped upon the inner or concave 
surface of a hollow cylinder, whilst the 
lower part is wrapped upon the outer or 
convex circumference of the same cylin- 
der, thus generating the thread of a fe- 
male screw within the cylinder, and a 
male screw without it ; and if the plane 
C be then wrapped upon the convex sur- 
"^^1^ face of a solid cylinder just fitting the in- 
side or concave surface of the above-mentioned hollow cylin- 

* It will be seen by reference to equation (301), that the working pressure 
P 2 depends for its amount, not upon the actual inclinations i x i* of the threads, 
but on the difference of their inclinations ; so that its amount may be enor- 
mously increased by making the threads nearly of the same inclination. Thus, 

\ t> t> cos - 'i cos - L<1 
neglecting friction, we have, by equation (301), r 2 =ri — ; — : -r- ; which 

SID.. yl\ l*xj 

expression becomes exceedingly great when ii nearly equals * 2 . 




VARIABLE INCLINATION OF THE THREAD. 



333 



der, and the plane A upon a concave cylindrical surface just 
capable of receiving and adapting itself to the outside or 
convex surface of that cylinder, the male screw thus genera- 
ted adapting itself to the thread of the screw within the hol- 
low cylinder, and the female screw to the thread of that 
without it ; if, moreover, the female screw last mentioned 
be fixed, and the solid male screw be free to traverse in the 
direction of its length, but be prevented turning upon its 
axis by the intervention of a guide ; if, lastly, a moving pres- 
sure or power be applied to turn the hollow screw, and a re- 
sistance be opposed to the longitudinal motion of the solid 
screw which is received into it ; then the combination will 
be obtained, which is represented in the preceding engraving, 
and which is well known as Mr. Hunter's screw, having been 
first described by that gentleman in the seventeenth volume 
of the Philosophical Transactions. 

The theory of this screw is identical with that of the pre- 
ceding, the relation of its driving and working pressures is 
determined by equation (301), and its modulus by equation 
(302). 



The Theory of the Screw with a Square Thread in re- 
ference TO THE VARIABLE INCLINATION OF THE THREAD AT 
DIFFERENT DISTANCES FROM THE AxiS. 



255. In the preceding investigation, the inclined plane 
which, being wound upon the cylinder, generates the thread 
of the screw, has been imagined to be an exceedingly thin 
sheet, on which hypothesis every point in the thread may be 
conceived to be situated at the same distance from the axis 
of the screw ; and it is on this supposition that the relation 
between the driving and working pressure in the screw and 
its modulus have been determined. 





Let us now consider the actual case in which the thread 



334: VARIABLE INCLINATION OF THE THBEAD. 

of the screw is of finite thickness, and different elements of 
it situated at different distance from its axis. 

Let mb represent a portion of the square thread of a screw, 
in which form of thread a line &e, drawn from any point b on 
the outer edge of the thread perpendicular to the axis ef, 
touches the thread throughout its whole depth bd. Let AC 
represent a plane perpendicular to its axis, and af the pro- 
jection of be upon tins plane. Take p any point in bd. and 
let q be the projection of p. Let ep=r, mean radius of 
thread =Ii, inclination of that helix of the thread whose 
radius is R* = I, inclination of the helix passing through p=i. 
whole depth of thread = 2D, distance between threads (or 
pitch) of screw = L. Now, since the helix passing through 
p may be considered to be generated by the enwrapping of 
an inclined plane whose inclination is i upon a cylinder 
whose radius is ?', the base of which inclined plane will then 
become the arc tq % we hnvepq—tq . tan. i. But, if the angle 
A/a be increased to 2*-, j?q will become equal to the com- 
mon distance L between the threads of the screw, and tq 
will become a complete circle, whose radius is r ; therefore 
L=2cr?> tan. t, and this being true for all values of ?•, there- 
fore L=2 < !rR tan. I. Equating the second members of these 
equations, and solving in respect to tan. t, 

E tan. I , . 

tan. i= (313). 

From which expression it appears, that the inclination of the 
thread of a square screw increases rapidly as we recede from 
its edge and approach its axis, and would become a right 
angle if the thread penetrated as far as the axis. Consider- 
ing, therefore, the thread of the screw as made up of an in- 
finite number of helices, the modulus of each one of which 
is determined by equation (312), in terms of its correspond- 
ing inclination i, it becomes a question of much practical im- 
portance to determine, if the screw act upon the resistance 
at one point only of its thread, at what distance from its axis 
that point should be situated, and if its pressure be applied 
at all the different points of the depth of its thread, as is 
commonly the case, to determine how far the conditions of 
its action are influenced by the different inclinations of the 
thread at these different depths. 

* This may be called the mean helix of the thread. The term helix is here 
taken to represent any spiral line drawn upon the surface of the thread; the 
distance of every point in which, from the axis of the screw, is the same. 



VARIABLE INCLINATION OF THE THREAD. 335 

We shall omit the discussion of the former case, and pro- 
ceed to the latter. 

Let P 2 represent the pressure parallel to its axis which is 
to be overcome by the action of the screw. Now it is evi- 
dent that the pressure thus produced upon the thread of the 
screw is the same as though the whole central portion of it 
within the thread were removed, or as though the whole 
pressure P 2 were applied to a ring whose thickness is As or 
2D. Now the area of this ring is represented by if j^R + D) 5 
— (R— D) 2 } , or by 4^RD. So that the pressure of r 2 , upon 

P 

every square unit of it, is represented by p' - . Let Ar 

represent the exceedingly small thickness of such a ring 
whose radius is r, and which may therefore be conceived to 
represent the termination of the exceedingly thin cylindrical 
surface passing through the point p ; the area of this ring is 
then represented by 2irr&r, and therefore the pressure upon 

it by ■ '' '' VJ ^ , or by t~. Now this is evidently the 

pressure sustained by that elementary portion of the thread 
which passes through p, whose thickness is Ar, and which 
may be conceived to be generated by the enwrapping of a 
thin plane, whose inclination is t, upon a cylinder whose ra- 
dius is r ; whence it follows (by equation 311) that the ele- 
mentary pressure aP i? which must be applied to the arm of 
the screw to overcome this portion of the resistance P 2 , thus 
applied parallel to the axis upon an element of the thread, 
is represented by 

a -d /P^ArV (r \ ( sin. (i -f <p/) cos. <p_ „o ) 

AP -= ( mwl U I cos.( l+ 9 1+ 9,) +V an ^ ! ; 

whence, passing to the limit and integrating, we have 

p -JL. r\ si "-('+p.) cos -y | hr tan „ I dr 

r '-2ED«./ \ cos. 0+9,+?,) +it % J 

R-D 

Now 
sin. (« + q>,) cos. <p 3 tan. i+tan. <p, 

cos. (« + <p, + <p 3 ) ~~ 1— tan. <p t tan. <p 3 — tan. » (tan. 9, + tan. <p t ) 

_ tan. i + tan. <p, 

""(1-tan. <p, tan. <p,){l-tan. ,tan. fa + 9,)} ^^ 9, + tan ' ' 



330 THE svur.w w nil A TBIAHOULAB THBEAB. 

4- tan. (^,-f9«) tan. V Neglecting dimensions of tan. 9, and 
tan. p, above the first*, 

R + D 

•' 1>i= 2iu>;/ « (tan - ^ +tan * ,+tan - (?i4 " 9s) ton - v+ 

R-D 

far tan. 9,} dr (314). 

Substituting in this expression for tan. 1 its value (equation 
313), it becomes 

R + D 

P l =g S ^T{r'tan.(p 1 +Brtan.I+E , taiL i Itan.(<p l +9,)+ 

R-D 

|prtan.9 a }r7;\ 
Integrating and reducing, 



P 1 ==i^jtan.I+(l+4j f )tan.9H-i(^)tan.9,+ 

tan. , Itan.(o 1 + 9,)| (315); 

whence we obtain by (equation 121) for the modulus, 

U 1 =L T a jl+|(l + ig)tan.9 1 -f|(^)tan.9 a + 

tan.'I tan. (9,4-93) j cot. 1 1 (316). 

256. "Whence it follows that the best inclination of the 
thread, in respect to the economy of power in the use of 
the square screw, is that which satisfies the equation 



tan, 



i=i( 1+ *gH* + *(sK* 



tau. (9! + 9,) 



The inclination of thread of a square screw rarely exceeds 
T°, so that the term tan. a I tan. (2, + 9,) rarely exceeds '015 
tan. (^~~\\ and may therefore be neglected, as compared 

* The integration is readily effected without this omission ; and it might be 
desirable so to effect it where the theory of wooden screws is under discussion, 
the limiting angle of resistance being, in respect to such screws, considerable. 
T!is length and complication of the resulting expression baa caused the omis- 
sion of i*. in the text. 



THE BEAM OF THE STEAM ENGINE. 



337 



with the other terms of the expression ; as also may the 
term i( — J tan. <p 1? since the depth 2D of a square screw 

being usually made equal to about -Jth of the diameter, this 
term does not commonly exceed T ^ tan. <pj. 

Omitting these terms, observing that Jj=2*~R tan. I, and 
eliminating tan. I, 



.-i^+Rtan.^+fptan.^l 



a 12* 



( 2-tt ) 

11,=^ l + r (K tan. 9, +|p tan. <p 2 ) V 



(317). 

(318). 



The Beam of the Steam Engine. 

257. Let P„ P,, P 3 , P 4 represent the pressures applied by 
the piston rod, the crank rod, the air pump rod, and the cold 




water pump rod, to the beam of a steam engine ; and sup- 
pose the directions of all these pressures to be vertical.* 

Let the rods, by which the pressures P„ P 2 , P 3 , P 4 are 
applied to the beam, be moveable upon solid axes or gud- 
geons, whose centres are a, d, b, e, situated in the same 
straight line passing through the centre of the solid axis 
of the beam. 

Let p n p 2 , p„ p, represent the radii of these gudgeons, p the 
radius of the axis of the beam, and <p„ <p„ <p 3 , <p 4 , 9 the limit- 
ing angles of resistance of these axes respectively. Then, if 
the beam be supposed in the state bordering upon motion 

* A supposition which in no case deviates greatly from tlu> troth, and :my 
error in which may be neglected, inasmuch as it can only influence t lie results 
about to be obtained in as far as they have reference to the friction of the 
beam; so that any error in the retail must be of two dimensions, at least, in 
respect to the coefficient of friction and the small angle by which any pressors 
deviates from a vertical direction. 

22 



33S THE BEAM OF THE 6TEAM ENGINE. 

by the preponderance of P„ each gudgeon or axis being 
upon the point of turning on its bearings, the directions 01 
the pressures P„ P 3 , P 3 , P 4 , P, will not be through the cen- 
tres of their corresponding axes, but separated from them by 
perpendicular distances severally represented by p, sin. <p„ p, 
sin. 9 9 , p s sin. <p„ p 4 sin. 9 4 , and p sin. 9, which distances, being 
perpendicular to the directions of the pressures, are all 
measured horizontally. 

Moreover, it is evident that the direction of the pressure 
P, is on that side of the centre a of its axis which is nearest 
to the centre of the beam, since the influence of the friction 
of the axis a is to diminish the effect of that pressure to turn 
the beam. And for a like reason it is evident that the 
directions of the pressures P 2 , P 3 , P 4 are farther from the 
centre of the beam than the centres of their several axes, 
since the effect of the friction is, in respect to each of these 
pressures, to increase the resistance which it opposes to the 
rotation of the beam ; moreover, that the resistance P upon 
the axis of the beam has its direction upon the same side of 
the centre C as P„ since it is equal and opposite to the 
resultant pressure upon the beam, and that resultant would, 
by itself, turn the beam in the same direction as P, turns it. 
Let now a 1 =Ca, a^=Cd, a s =Cb, a A =Ce. Draw the hori- 
zontal line offCg, and let the angle aCf=K Let, moreover, 
W be taken to represent the weight of the beam, supposed 
to act through the centre of its axis. Then since P n P 2 , P s , 
P 4 , "W, R are pressures in equilibrium, we have, by the 
principle of the equality of moments, taking o as the point 

from which the moments are measured, P t . qf=Y 2 . og+ 
F,.oh+F 4 „<>k+W.oC. 

Now of=Cf—Co=a 1 cos. Q— p, sin. 9,— p sin. 9, og=Cg+ 
Co=a, cos. d + p 2 sin. 9, + p sin. 9, oh=Ch—Co=a i cos. d-h 
p 8 sin. <p 3 — p sin. 9, ok=Ck+Co=a A cos. d + p 4 sin. <p 4 H-p sin. 9. 

.•.PJtfj cos. d— (p x sin. <p, + p sin. 9)} = 

P 2 \a 3 cos. 6 + (p 9 sin. 9 2 + p sin. 9)} + 
P 1 ia i cos.a + (p,sin.9 1 — psin.9)|+ L. . .(319). 

P 4 \a t cos. 6 + (p 4 sin. 9 4 + p sin. 9)} + Wp sin. 9 

Multiplying this equation by 6, observing that afi repre- 
sents the space described by the point of application of P„ 
so that Y^afi represents the work U, of P, ; and similarly 
that P,a 3 d represents the work U 2 of P 2 , P 8 « 3 d, that LT 3 of P„ 
and P A aJ, that U 4 of P 4 , also that afi represents the space S, 



THE BE A3! OF THE STEAM ENGINE. 



339 



described by the extremity of the piston rod very nearly ; 
we have 

Ui J C0S . (! -(e>^-+p sin - ? )[= 

uJco 9 .d+( p ' sin -\ +psin - 9 )[ + 



!aWt<+ Mh.^ f rin.t | 



(320), 



[u,jcoM + ( ^ i "-^^ i ^ )j + WS,(l) si n., J 

which is the modulus of the beam. 

Its form will be greatly simplified if we assume cos. 0=1, 
since is small,* suppose the coefficient of friction at each 
axis to be the same, so that cp=9 1 =:9 2 =cp 3 =:9 it , and divide by 
the coefficient of U 15 omitting terms above the first dimen- 
sion in — sin. <p, &c. ; whence we obtain by reduction 



u. 



u 4 ji + (^ + E±b)*.,[ + ^(a*., 



(321). 



258. TAtf best position of the axis of the beam. 

Let a be taken to represent the length of the beam, and x 
the distance aC of the centre of its axis from the extremity 
to which the driving pressure is applied. 




* In practice the angle 6 never exceeds 20°, so that cos. 6 never differs from 
unity by more than '060307. The error, resulting from which difference, in 
the friction, estimated as above, must in all cases be inconsiderable. 



340 THB BEAM Oi IBB BHCAM ENGINE. 

Let the influence of the position of the axis on the 
economy of the work necessary to open the valves, to work 
the air-pump, and to overcome the friction produced by the 
weight of the axis, be neglected ; and let it be assumed to 
be that, by which a given amount of work U, may be 
yielded per stroke upon the crank rod, by the least possible 
amount U, of work done upon the piston rod. If, then, in 
equation (321), we assume the three last terms of the second 
member to be represented by A, and observe that a l in that 
equation is represented by a?, and # 3 by a — x, we shall 
obtain 

The best position of the axis is determined by that value 
of x which renders this function a minimum; which value 
of a? is represented by the equation 



x= 1 _ (322.) 

V+f/ 

If p,>p 1 , then! 2 ) >1 and x<.ia; in this case, there- 
fore, the axis is to be placed nearer to the driving than to 
the working end of the beam. If p„<p„ the axis is to be 
fixed nearer to the working than to the driving end of the 
beam. 

259. It has already been shown (Art. 168.), that a 
machine working, like the beam of a steam engine, under 
two given pressures about a fixed axis, is worked with the 
greatest economy of power when both these pressures are 
applied on the same side of the axis. This principle is 
manifestly violated in the beam engine; it is observed in 
the engine w T orked by Crowther's parallel motion,* and in 
the marine engines recently introduced by Messrs. Seaward, 
and known as the Gorgon engines. It is difficult indeed to 
defend the use of the beam on any other legitimate ground 
than this, that in 6ome degree it aids the fly-wheel to 
equalise the revolution of the crank arm,f an explanation 

* As used in the mining districts of the north of England. 

\ The reader is referred to an admirable discussion of the equalising power 
of the beam, by M. Coriolis, contained in the thirteenth volume of the Journal 
de VEcole Poly technique. 



THE CEANK. 



341 



which does not extend to its use in pumping engines, 
where, nevertheless, it retains its place ; adding to the 
expense of construction, and, by its weight, greatly increas- 
ing the prejudicial resistances opposed to the motion of the 
engine. 



The Crank. 

260. The modulus of the crank, the direction of the resist- 
ance being parallel to that of the driving pressures. 



Let CD represent the arm of the crank, and AD the con- 
necting rod. And to simplify the 
investigation, let the connecting 
rod be supposed always to retain 
its vertical position.* Suppose the 
weight of the crank arm CD, act- 
ing through its centre of gravity, 
to be resolved into two other 
weights (Art. 16), one of which W 2 
is applied at the centre C of its axis 
and the other at the centre c of 
the axis which unites it with the 
connecting rod. Let this latter 
weight, when added to the weight 
of the connecting rod, be repre- 
sented by W,. Let P 2 represent a 
pressure opposed to the revolution 
of the crank, which would at any 
instant be just sufficient to balance 
the driving pressure P, transmitted through the connecting 
rod ; and to simplify the investigation, let us suppose the 
direction of the pressure P 2 to be vertical and downwards. 

Let Cc=a, CA l =a l , CA 2 =& 2 , cCW 2 =d, radii of axes C 
and <?=p„ p a , lim. /s of resistance =<p„ <p 9 , W= whole weight 
of crank arm and connecting rod=W 1 -f-W 2 . 

Since the crank arm is in the state bordering upon 
motion, the perpendicular distance of the direction of the 
resistance upon its axis C from the centre of that axis, is 




* Any error resulting from this hypothesis affecting the conditions of the 
question only in as far as the friction is concerned, and being of two dimon- 
sions at least in terms of the coefficient of friction and the small angular devi- 
ation of the connecting rod from the vertical. 



342 



THE CKANK. 



represented by p, sin. <p, (Art. 153.). The resistance is i,lso 
equal to P a ± (P, + W) ; I*! being supposed greater than P, + W ; 

and the sign ± being taken 
according as the direction of 
P, is downwards or upwards, 
or according as the crank arm 
is describing its descending or 
ascending arc. Whence it 
follows, that the moment of 
the resistance of the axis about 
its centre is represented by 
jP 1 ±(P, + W)i Pl sin. 9l . 
Now the pressures P x , P 2 , and 
the resistance of the axis, are 
pressures in equilibrium. 
Therefore, by the principle of 
the equality of moments, ob- 
serving that the driving pressure is represented by 'P l ±W li 
according as the arm is descending or ascending, 

(P,±W,) a,=P,«,+ SP,±(P, + W)S Pl sin. 9l . 

Since moreover the axis c, which unites the connecting 
rod and the crank arm, is upon the point of turning upon 
its bearings, the direction of the pressure P, is not through 
the centre of that axis, but distant from it by a quantity 
represented by p 2 sin. p 2 , which distance is to be measured 
on that side of the centre c which is nearest to C, since the 
friction diminishes the effect of P x to turn the crank arm. 




a,=a sin. e— p 2 sm. <p a 



(323). 



Substituting this value of a x in the preceding equation, 
(P.iW,) (a sin. d- Pa sin. <p,)=P.« 9 + [P l ±(P i + 

W)} Pl sin. <p> (324). 

Transposing and reducing 

P, \a sin. &— p 3 sin. <p 2 — p x sin. <p t } =P a {a 9 ±p 1 sin. pj ± 
Wp, sin. 9 1 =pW 1 (a sin. d — p 2 sin. 9 9 ); 

which is the relation between P, and P 2 in their state bor- 
dering upon motion. Now if Ad represent an exceedingly 
small angle described by the crank arm, a^& will represent 
the space through which the resistance P 2 is overcome 
whilst that angle is described, and P 2 # 2 A<) will represent the 



THE CRANK. 343 

increment aU 2 of the work yielded by the crank whilst that 
small angle is described. Multiplying the above equation 
by a 2 A0, we have 

P,# 2 j# sin. 8— p 2 sin. cp 2 — Pl sin. cp 1 \A$=\a< i ±p l sin. (pjATJ 9 ± 

Waj t sin. <p A0=F "W^ (a sin. 0— p 2 sin. ? 2 )A0 .... (325). 

whence passing to the limit, integrating from 0=0 to d= 
tf — 0, and dividing by # 2 

P, |2acos.0— (^— 20)(p 2 sin.<p 2 + p 1 sin.(p 1 )} = | 1± Asin.^ !■ U a ± 

W^— 20) Pl sin. 9 i q=W 1 {2acos.e— p 2 (*-20) sin. <p 9 } . . (326). 

Now, let it be observed that 2a cos. represents the pro- 
jection of the path of the point c upon the vertical direction 
of P 1? whilst the arm revolves Lotween the positions and 
if— ; so that Y x 2a cos. represents (Art. 52.) the work 
U 1 done by P x upon the crank whilst the arm passes from 
one of these positions to the other, or whilst the work U 2 is 

yielded by the crank. "Whence it follows that T 1 =^ 2 sec. 0. 

Substituting this value of P x , and reducing we obtain 

U 1 jl_g-0)sec.0gsin., 2 + ^sin. 9l )} = 

| 1±£ sin. <p, J U 2 ±W (tf-20) Pl sin. cp^W, {2a cos. ©- 

Pa (<r-20) sin. <p 2 J (327). 

By which equation is determined the modulus of the crank 
in respect to the descending or ascending stroke, according 
as we take the upper or lower signs of the ambiguous terms. 
Adding these two values of the modulus together, and 
representing by \J, the whole work of P„ and by U 2 the 
whole work of P 2 , whilst the crank arm makes a complete 
revolution, also by u x the work of P 2 in the down stroke, 
and u 2 in the up stroke, we obtain 



U, j 1- (J-®)sec. e(£ sin. <?,+h s ;„. ?i j J = u, + 

(«•— «») r sin. <p, (328), 

which is the modulus of the crank in respect to a vertical 



344: THE CRANK. 

direction of the driving pressure and of the resistance, the 
arm being Btipposed in each half revolution, first, to receive 
the action of the driving pressure when at an inclination ot 
to the vertical, and to yield it when it has again attained 
the same inclination, so as to revolve under the action of 
the driving pressure through the angle * — 2®. 

In the double-acting engine, u 1 —u n = ; in the single-act- 
ing engine u x —0. The work expended by reason of the 
friction of the crank is therefore less in the latter engine than 
in the former, when the resistance P 2 is applied, as shown 
in the figure, on the side of the ascending arc. 

If the arm sustain the action of the driving pressure con- 
stantly r , 0=0, and the modulus becomes, for the double-act- 
ing engine, 

tT,{l_*(jeinM-jBin. ?1 )}=IT,; 

or, dividing by the co-efficient of- TJ l and neglecting dimen- 
sions above the first in sin. p l5 sin. <p 2 , 

U^ll+^^sin.^ + ^sin. 9 2 ) JU 2 .... (329). 

The modulus not involving the symbol "W" which repre- 
sents the weight of the crank, it is evident that so long as P, 
and P 2 are vertical and P, greater than P 2 -h"W, the economy 
of power in the use of the crank is not at all influenced by 
its weight and that of the connecting rod, the friction being 
upon the whole as much diminished by reason of that weight 
in the ascending stroke as it is increased by it in the descend- 
ing stroke. 

It is evident, moreover, that if the friction produced by 
the weight of the crank be neglected, the modulus above de- 
duced, for the case in which the directions of the pressures 
P, and P 2 are vertical, applies to every case in which the 
directions of those pressures are parallel. 

The condition P,>P 2 -|-W evidently obtains in every other 
position of the crank arm, if it obtain in the horizontal position. 

Now, in this position, P 2 =— P x , if we neglect friction. The 

required condition obtains, therefore, if P x >— Pj + TV. To 

satisfy this condition, « 2 must be greater than «, or the 
resistance be applied at a perpendicular distance from the 



THE DEAD POINT IN THE CRANK. 345 

axis greater than the length of crank arm, and so mnch 

greater, that Y 1 ll ) may exceed W. These conditions 

commonly obtain in the practical application of the crank. 

261. Should it, however, be required to determine the mo- 
dulus in the case in which P, is not, in every position of the 
arm, greater than P 2 + W, let it be observed, that this condi- 
tion does not affect the determination of the modulus (equa- 
tion 327) in respect to the descending, but only the ascend- 
ing stroke ; there being a certain position of the arm as it 
ascends in which the resultant pressure upon the axis repre- 
sented by the formula {P 1 — (P 2 -\-W)\ , passing through zero, 
is afterwards represented by {(P 2 + W)— P x | ; and when the 
arm has still further ascended so as to be again inclined to the 
vertical at the same angle, passes again through zero, and is 
again represented by the same formula as before. The value 
of this angle may be determined by substituting P 2 for 
P a -f-W in equation (324), and solving that equation in re- 
spect to 0. Let it be represented by X ; let equation (325) 
be integrated in respect to the ascending stroke from 0=0 
to 0=^, the work of P 2 through this angle being represented 
by u 1 ; let the signs of all the terms involving p x sin. <p x then 
be changed, which is equivalent to changing the formula re- 
presenting the pressure upon the axis from {Pj — (P 9 + ~W)| 
to $(P 2 + W)— PjJ; and let the equation then be integrated 



it 



from 0=0, to 0=-, the work of P, through this angle being re- 

presented by u^\ 2( / w 1 -t-w 2 ) will then represent the whole 
work U 2 done by P 2 in the ascending arc. To determine 
this sum, divide the first integral by the co-efficient of u iy 
and the second by that of u„ add the resulting equations, 
and multiply their sum by 2 ; the modulus in respect to the 
ascending arc will then be determined ; and if it be added 
to the modulus in respect to the descending arc, the modu- 
lus in respect to an entire revolution will be known. 



The Dead Points in the Crank. 

362. If equation (324) be solved in respect to P x it be- 
comes 



346 THE DOUBLE CRANK. 

p = p i «,±P,sin. <p, ) 

1 3 ( a sin. 0— p 2 sin. <p a — p x sin. <p, ) 

Wp, sin.g^— "WXgsin.fl— p a sin. <p 3 ) 
a sin. ^ — p a sin. 9 3 — p, sin. 9 X 

In that position of the arm, therefore, in which 

Bin<= P,rin.9,+P,sin.9, _ _ 



a 



the driving pressure P 2 necessary to overcome any given re- 
sistance P„ opposed to the revolution of the crank, assumes 
an infinite value. This position from which no finite pres- 
sure acting in the direction of the length of the connecting 
rod is sufficient to move the arm, when it is at rest in that 
position, is called its dead point. 

Since there are four values of 0, which satisfy equation 
(330) two in the descending and two in the ascending semi- 
revolution of the arm, there are, on the whole, four dead 
points of the crank.* The value of Pj being, however, in all 
cases exceedingly great between the two highest and the two 
lowest of these positions, every position between the two 
former and the two latter, and for some distance on either 
side of these limits, is practically a dead point. 



The Double Crank. 

263. To this crank, when applied to the steam engine, are 
affixed upon the same solid shaft, two arms at right angles 
to one another, each of which sustains the pressure of the 
steam in a separate cylinder of the engine, which pressure is 
transmitted to it, from the piston rod, by the intervention of 
a beam and connecting rod as in the marine engine, or a 
guide and connecting rod as in the locomotive engine. 



* It has been customary to reckon theoretically only two dead points of the 
crank, one in its highest and the other in its lowest position. Every practical 
man is acquainted with the fallacy of this conclusion. 



THE DOUBLE CRAIs T K. 



347 




In either case, the connecting rods 
may be supposed to remain con- 
stantly parallel to themselves, and 
the pressures applied to them in 
different planes to act in the same 
% plane,* without materially affecting 
the results about to be deduced.f 

Let the two arms of the crank be 
supposed to be of the same length a ; 
let the same driving pressure P, be 
supposed to be applied to each ; and 
let the same notation be adopted in 
other respects as was used in the 
case of the crank with a single arm; 
and, first, let us consider the case 
represented in fig. 1, in which both 
arms of the crank are upon the same 
side of the centre C. 



Let the angle W 1 CB=« 



therefore W X CE— - + 6 : whence 



it follows by precisely the same reasoning as in Art. 260., 
that the perpendicular upon the direction of the driving 
pressure applied by the connecting rod AB is represented 
(see equation 323) by a sin. 4— p 2 sin. <p 2 , and the per- 
pendicular upon the pressure applied by the rod ED by 

a sin.(-+ 0] — p a sin. <p 9 , or a cos. d— p 2 sin. <p a . Let now a x 

be taken to represent the perpendicular distance from the 
axis C, at which a single pressure, equal to 2P X , must be ap- 
plied, so as to produce the same effect to turn the crank as 
is produced by the two pressures actually applied to it by 
the two connecting rods ; then, by the principle of the equa- 
lity of moments, 

2P l a 1 =P 1 (tf sin.d— p 9 sin. cp^ + P^a cos. &— p 2 sin. 9,) ; 
.\ a x -=\a (sin. 6 -f cos. d)— p 9 sin. <p a ; 



* This principle will be more fully discussed by a reference to the theory of 
statical couples. (See Pritchard on Statical Couples.) 

f The relative dimensions of the crank arm and connecting rod are here sup- 
posed to be those usually given to these parts of the engine ; the supposition 
does not obtain in the case of a short connecting rod. 



348 THE DOUBLE CRANK. 

;. a t = — = I sin. 6 cos. j + cos.e sin. -I — p a sm. p,= 

which expression becomes identical with the value of a x , de- 
termined by equation (323), if in the latter equation a be 

Teplaced by -— , and 6 by 6 -f -. Whence it follows that the 

conditions of the equilibrium of the double crank in the 
state bordering upon motion, and therefore the form of the 
modulus, are, whilst both arms are on the same side of the 
centre, precisely the same as those of the single crank, the 
direction of whose arm bisects the right angle BCE, and 
the length of whose arm equals the length of either arm of 
the double crank divided by |/2. 

Now, if 6 X be taken to represent the inclination WjCF of 
this imaginary arm to "Wfi, both arms will be found on the 

same side of the centre, from that position in which 6 X = - 
to that in which it equals ( if — -). If, therefore, we substi- 
tute - for 0, in equations (326), and for <z, — — , and add these 

equations together, the symbol 2 U 2 in the resulting equa- 
tion will represent the whole work yielded by the working 
pressure, whilst both arms remain on the same side of the 
centre, in the ascending and the descending arcs. We thus 
obtain, representing the sum of the driving pressures upon 
the two arms by P l5 

2P 1 {a-!(p J sin. ?2 + PlS in.9,)S=2U, (331).* 

Throughout the remaining two quadrants of the revolution 
of the crank, the directions of the two equal and parallel 
pressures applied to it through the connecting rods being 
opposite, the resultant pressure upon the axis is represented 
by (P 2 + W), instead of JP 1 ±(P 2 + W)S ; whilst, in other 
respects, the conditions of the equilibrium of the state bor 

* Whewell's Mechanics, p. 25. 



THE DOUBLE CRANK. 



849 



Fig. 2. 




dermg upon motion remain the same as before ; that is, the 
same as though the pressure P x were 
applied to an imaginary arm, whose 

length is — 7=, and whose position co- 

incides with CF. Now, referring to 
equation (324), it is apparent that 
this condition will be satisfied if, in 
that equation, the ambiguous sign of 
(P 9 +W) be suppressed, and the 
value of Pj in the second member, 
which is multiplied by p, sin. <p l5 be 
assumed =0 ; by which assumption 
the term — p 1 sin. <p, will be made to 
disappear from the left-hand member 
of equation (325), and the ambiguous 
signs which affect the first and second 
terms of the right-hand member will become positive. Now, 
these substitutions being made, and the equation being then 

integrated, first, between the limits and -, and then be- 

tween the limits — and at, the symbol U 2 in it will evidently 

represent the work done during each of those portions of a 
semi-revolution of the imaginary arm in which the two real 
arms of the crank are not on the same side of the centre. 
Moreover, the integral of that equation between the limits 

If 

and jj is evidently the same with its integral between the 

3* 
limits -7- and *. Taking, therefore, twice the former inte- 
gral, we have 

2Pa 1 71 ( 1_cos *i) ~ i pa sin * 9i I = ] a * + pi sin * 9l I 

if C a l if\ if ) 

2U 2 + 2 W« a Px sin. ?J2¥^ S \ — |l~cos.j] -j p a sin. <p, J . 

Dividing this equation by (# a + p l sin. <p,), or by a t 
( 1+ — ! sin. (p, h and neglecting terms above the first dimen- 
sion in sin. p, and sin. <p a , 



350 THE DOUBLE CRANK. 

2P,j^(l-coB.g(l r j8in. 9l )-jP,Bin.9, }=2U,+ 
;Wp l8 in. 9l T2W l |-^(l-co8.j) ( 1- Jsin. 9l )- 

in which equation 2U, represents the work done in the 
descending or ascending arcs of the imaginary arm, accord- 
ing as the ambiguous sign is taken positively or negatively. 
Taking, therefore, the sum of the two values of the equation 
given by the ambiguous sign, and representing by 4U, the 
whole work clone in the descending and ascending arcs, dur- 
ing those portions of each complete revolution when both of 
the arms are not on the same side of the centre, we have 

* p .{-^( 1 -««-i)(i-5^^)-i ,, ' 8ilL9 ' 1 = 

4TJ a -f "Wflrpj sin. cp l ; 

* 1 

or, observing that cos. t = ~~ i=<> 

2P, | a (i/2-l)-a( |/2-l) £ sin. <p - \ P, sin. <p, j = 
4TT a +'WVp 1 sin. (jv 

Adding this equation to equation (331), and representing by 
U 3 the entire work yielded during a complete revolution of 
the imaginary arm, 

2P, | a |/2 - a ( |/2 - 1) £ sin. tp, — | (2 Pa sin.?, + p 1 sin.?,) | 

=U a +W^p, sin.?,. 

But if U, represent the whole work done by the driving 
pressures at each revolution of the imaginary arm, then 

4—— P 1 =U l . Since 2 — j= is the projection of the space 



THE CRANK GUIDE. 351 

described by the extremity of the arm during the ascending 
and descending strokes respectively, therefore 2P t = — y=. 
Substituting this value for 2P X , 

IT, | l-^=i£ ™. ft-jjgf* sin. ,,+ t sin. I = 

U a +'W*p 1 Bin.«p 1 (332), 

which is the modulus of the double crank, the directions of 
the driving pressure and the resistance being both supposed 
vertical ; or if the friction resulting from the weight of the 
crank be neglected, and W be therefore assumed =0, then 
does the above equation represent the modulus of the 
double crank, whatever may be the direction of the driving 
pressure, provided that the direction of the resistance be 
parallel to it. Dividing by the coefficient of Uj, and 
neglecting terms of more than one dimension in sin. <p x and 
sin. <p a , 

__ „ j H 4/2—1 Pl . * /2p 3 . 
■£■ sin. 9» ) I + W^sin.9, (333). 



The Crank Guide. 

264. In some of the most important applications of the 
steam engine, the crank is made to receive its continuous 
rotatory motion, from the alternating rectilinear motion of 
the piston rod, directly through the connecting rod of the 
crank, without the intervention of the beam or parallel 
motion ; the connecting rod being in this case jointed at one 
extremity, to the extremity of the piston rod, and the oblique 
pressure upon it which results from this connexion being 
sustained by the intervention of a cross piece fixed upon it, 
and moving between lateral guides.* 

* This contrivance is that well known as applied to the locomotive carriage 



352 



THE CRANK GUIDE. 




Let the length CD of the connecting rod be represented 
by b, and that BD of the crank arm by a, and let r, and P, 
in the above figure be taken respectively, to represent the 
pressure upon the piston rod of the engine and the connect- 
ing rod of the crank, and RS to represent the direction of 
the resistance of the guide in the state bordering; upon 
motion by the excess of the driving pressure P,. Then is 
RS inclined to a perpendicular to the direction of the guides, 
or of the motion of the piston rod, at an angle equal to the 
limiting angle of resistance (Art. 141) of the surfaces of con- 
tact of the guides. 

Since, moreover, P„ P 2 , R are pressures in equilibrium, 

P 2 sin. P,CS 



••^-"sin.P^S* 
Let /BCD=d ; limiting angle of resistance of guide =9 
therefore, P.CS^-?, P 2 CS=J+«p-4 ; 



sin.g- ? ) 



cos. 9 



^9~(^ — <P) 



COS. 



■?) 



(334). 



Let BD = a, CD = 5, and DBC = d 15 and assume P, to 
remain constant, P, being made to vary according to the 
conditions of the state bordering upon motion ; 

.-. aU^P, . aAC= -P, . aBC= -P x . a (a cos. ^ + 5 cos. 6)=z 
P 2 sec. <p cos. (6— <p) (a sin. 6 l A6 i -f- b sin. 0A0) ; 

aU 9 =— P 3 (ABC)cos.^=P a (asin.^Ad 1 + 5sin.flA4)cos.^; 

IT 

.\\J 1 =T t sec.(p\a I sinJ 1 cos.(^-~<p)^ l -fJ / sinJcos.(d— <p)<$}. 



THE FLY-WHEEL. 



353 



U a =P a {a J Bin. 6 l cos. bdfi x + l J sin. 6 cos. 6<$\ = 
o o 

The second integral in each of these equations vanishes 



a . 



between the prescribed limits ; also sin. 6 = - sin. d, ; there- 



fore cos. 6 = (1— — sin. \)*; 



;. V^T.afsm. \ cos. ^ X -V % af{l - ^ sin. •*,)* sin. &,d& x = 



7T 7T 

TT^P^sec. <p / sin. ^ cos. (6— (p)d^ 1 =7^a I sin. ^ cos. dd^-f 
o o 

7T 7T 

P a atan. 9 / sin. 6 sin. ^ 1 =U a -f-P 9 ^-tan. <p / ism. \ ^ x =. 



TT a 4-P 2 i^tan.<p; 

whence eliminating P 2 and reducing, we obtain 

* tan. 9 



u,=u j 



i + 



which is the modulus of the crank guide. 



(335), 



The Fly-Wheel. 
265. TAtf angular velocity of the fly-wheel. 

Let Pj be taken to represent a constant pressure applied 
through the connecting rod to the arm of the crank of a 

* Church's Diff and Int. Cal. Art. 199. 
23 



864 



THE FLY-WHEEL. 




-jir ._-- 



steam engine : suppose the direction of this pressure to 
remain always parallel to itself, and let P 2 represent a cew- 
sfcm£ resistance opposed to the revolution of the axis which 
carries the fly-wheel, by the useful work done and the pre- 
judicial resistances interposed between the axis of the 
fly-wheel and the working points of the machine. 

Let the angle ACB=d, CB=a, CP 2 r=« 2 . 

Now the projection, upon the direction of P„ of the path 
of its point of application B to the crank arm, whilst that 
arm describes the angle ACB, is AM, therefore (Art. 52.), 
the work done by T 1 upon the crank, whilst this angle is 
described, is represented by P, . AM, or by P, a vers. d. 
And whilst the crank arm revolves through the angle #, the 
resistance P 2 is overcome through the arc of a circle sub- 
tended by the same angle d, but whose radius is a„ or 
through a space represented by a£. So that, neglecting the 
friction of the crank itself, the work expended upon the 
resistances opposed to its motion is represented by P 2 #A and 
the excess of the work done upon it through the angle ACB 
by the moving power, over that expended during the same 
period upon the resistances, is represented by 

P/zvers.d-P^ (336). 

Now 2<zP, represents the work done by the moving pressure 
P, during each effective stroke of the piston, and 2*-a 2 P 2 the 
work expended upon the resistance during each revolution 
of the fly-wheel ; so that if m represent the number of 
strokes made by the piston whilst the fly-wheel makes one 



THE FLY-WHEEL. 355 

revolution, and if the engine be conceived to have attained 
its state of uniform or steady action (Art. 146.), then 
2maP 1 =2*u,P a , 

.'.af^aP, (337). 

Eliminating from equation (336) the value of # 2 P 3 deter- 
mined by this equation, we obtain for the excess of the work 
done by the power (whilst the angle d is described by the 
crank arm), over that expended upon the resistance, the 
expression 

P^jverB.a-^1 (338). 

But this excess is equal to the whole work which has been 
accumulating in the different moving parts of the machine, 
whilst the angle 6 is described by the arm of the crank (Art. 
145). Now, let the whole of this work be conceived to have 
been accumulated in the fly-wheel, that wheel being pro- 
posed to be constructed of such dimensions as sufficiently to 
equalise the motion, even if no work accumulated at the 
same time in other portions of the machinery (see Art. 150.), 
or if the weights of the other moving elements, or their 
velocities, were comparatively so small as to cause the work 
accumulated in them to be exceedingly small as compared 
with the work accumulated during the same period in the 
fly-wheel. Now, if I represent the moment of inertia of the 
fly-wheel, p the weight of a cubic foot of its material, a t its 
angular velocity when the crank arm was in the position 
CA, and a its angular velocity when the crank arm has 

Ja 

passed into the position CB ; then will \ — (a 2 — a*) represent 

the work accumulated in it (Art. 75.) between these two 
positions of the crank arm, so that 

.^■JEjF{«*4-$j (339). 



266. The positions of greatest and least angular velocity of 
the fly-wheel. 

If we conceive the engine to have acquired its state of 
steady or uniform motion, the aggregate work done by the 



356 



THE FLY-WUEJEL. 




power being equal to that expended upon the resistances, 
then will the angular velocity of the fly-wheel return to the 
same value whenever the wheel returns to the same position ; 
so that the value of a 1 in equation (339) is a constant, and 
the value of a a function of & ; a assumes, therefore, its mini- 
mum and maximum values with this function of 0, or it is a 



minimum when 



<ft 



0, 



da . -, (Pa ^ A 

W = > and ^ <0 - 



and-^->0, 
da' 



and a maximum when 



772, 



But -^7-= sin. 6 — 

dA * 



and 



w =cos -*> 



therefore -^-=0, when 



m 



sm. e=- 



(340.) 



!Now this equation is evidently satisfied by two values of 
0, one of which is the supplement of the other, so that if -n 
represent the one, then will (#— *i) represent the other; 
which two values of 6 give opposite signs to the value cos. 
6 of the second differential co-efficient of a a , the one being 
positive or >0, and the latter negative or <0. The one 
value corresponds, therefore, to a minimum and the other 
to a maximum value of a. If, then, we take the angle ACB 



in 



in the preceding figure, such that its sine may equal — 

(equation 340), then will the position CB of the crank arm 
be that which corresponds to the minimum angular velocity 



THE FLY-WHEEL. 357 

of the fly-wheel ; and if we make the angle ACE equal to 
the supplement of ACB, then is CE the position of the 
crank arm, which corresponds to the maximum angular 
velocity of the fly-wheel. 

267. The greatest variation of the angular velocity of the 
fly-wheel. 

Let a 2 be taken to represent the least angular velocity of 
the fly-wheel, corresponding to the position CB of the crank 
arm, and a 3 its greatest angular velocity, corresponding tc 

1*1 

the position CE ; then does ~- (a 3 2 — a 2 2 ) represent the work 

accumulated in the fly-wheel between these positions, which 
accumulated work is equal to the excess of that done by the 
power over that expended upon the resistances whilst the 
crank arm revolves from the one position into the other, 
and is therefore represented by the difference of the values 
given to the formula (338) when the two values at— ^ and 
i, determined by equation (340), are substituted in it for 6. 
How this difference is represented by the formula 

r,a i vers. (*— *i)— vers. >]— r , 

( / 2-i\ ) 

or by P^ i 2 cos. *j— m 1 1 — —I t ; 

.•.|(a 3 '-0=P l « ) 2 cos. „-m(l - J) \ ; 

,^- a - 2 4fJ2co,,- m (l-5)( (341); 

in which equation n is taken (equation 340) to represent 

i it . m 

that angle whose sine is — . 



268. The dimensions of the fly-wheel, such that its angular 
velocity may at no period of a revolution deviate beyond 
prescribed limits from the mean. 

Let JN" be taken to represent the mean number of revo- 

N 
lutions made by the fly-wheel per minute; then will i^ 



358 THE FLY- WHEEL. 

represent the mean number of revolutions or parts of a 

revolution made by it per second, and fc^*, or — -, the 

mean space described per second by a point in the fly-wheel 
whose distance from the centre is unity, or the mean angular 
velocity of the fly-wheel. Now, let the dimensions of the 
fly wheel be supposed to be such as are sufficient to cause 
its angular velocity to deviate at no period of its revolution 

by more than -th from its mean value ; or such that the max- 



n 



imum value a s of its angular velocity may equal -^ I 1 + -I 
and that its minimum value a 3 may equal -^r-l 1 J ; then 

Substituting in equation (341), 



it 
30 



w- , -srJ» i ~ ,, H 1 -v)i 



Let H be taken to represent the horses' power of the 
engine, estimated at its driving point or piston ; then will 
33000H represent the number of units of work done per 
minute, upon the piston. But this number of units of work 
is also represented by %Nm . 2P 1 a ; since £Nm is the number 
of strokes made by the piston per minute, and 2P t a is the 
work done on the piston per stroke, 

;.2P<0=66OOOtJ=-. 

Substituting this value for 2P x a in the above equation, we 
obtain, by reduction, 

_ (66000.30V) ( /, 2>i\ ) Ha, fe%A ^ 

Let k be taken to represent the radius of gyration of the 
wheel, and M its volume ; then (Art. 80.) M/c a =I, therefore 
fjoM . & a =f/J. But m-M represents the weight of the wheel 
in lbs. ; let W represent its weight in tons ; therefore, 
uM=2240W. Substituting this value, and solving in 
respect to W, 



THE FLY-WHEEL. 359 



_ (66000.30^) ( I %n\ \ 



Wmtf 



Substituting their values for ne and g y and determining the 
numerical value of the co-efficient, 

W=86491 { | cos. ,- (l - 5) ) ** (343). 

If the influence of the work accumulated in the arms of 
the wheel be given in, for an increase of the equalising 
power beyond the prescribed limits, that accumulated in the 
neavy rim or ring which forms its periphery being alone 
taken into the account ;* then (Art. 86.) M.k* =I=2«bcR 
(R 2 +Jc 2 ), where b represents the thickness, c the depth, and 
E the mean radius of the rim. But by Guldinus's first 
property (Art. 38.), 2*5oK=M; therefore & 2 =(K 2 -bic 3 ). 
Substituting in equation (343) 

W=86491 j I cos. ■„ ( 1 - J) } W §^ f) .... (344). 

If the depth c of the rim be (as it usually is) small as 
compared with the mean radius of the wheel, Jc 2 may be 
neglected as compared with R 2 , the above equation then 
becomes 

W=86491 ) | cos. ,_(l -|) } *k • • • • (3«) 5 

by which equation the weight W in tons of a fly-wheel of a 
given mean radius R is determined, so that being applied to 
an engine of a given horse power Ii, making a given num- 
ber of revolutions per minute %N, it shall cause the angular 

velocity of that wheel not to vary by more than -th from its 

mean value. It is to be observed that the weight of the 
wheel varies inversely as the cube of the number of strokes 
made by the engine per minute, so that an engine making 
twice as many strokes as another of equal horse power, 

* If the section of each arm be supposed uniform and represented by «, and 
the arms be p in number, it is easily shown from Arts. 79., 81., that the 
momentum of inertia of each arm about its extremity is very nearly repre- 
sented by i«(R— ic) 3 , where c represents the depth of the rim ; so that the 

whole momentum of inertia of the arms is represented by ^k(R— £c) 8 , which 

8 
expression must be added to the momentum of the rim to determine the whole 
momentum I of the wheel. It appears, however, expedient to give the inertia 
of the arms to the equalising power of the wheel. 



360 THE FLY-WHEEL. 

would receive an equal steadiness of motion from a fly- 
wheel of one eighth the weight ; the mean radii of the 
wheels being the same. 

If, in equation (342), we substitute for I its value 2*bcR*, 
or 2tKIv 3 (representing by K the section be of the rim), and 
if we suppose the wheel to be formed of cast iron of mean 
quality, the weight of each cubic foot of which may be 
assumed to be 450 lb., we shall obtain by reduction 

E^6S5 2 ljlco,,-(l-^)j|| (3,6); 

by which equation is determined the mean radius R of a fly- 
wheel of cast iron of a given section K, which being applied 
to an engine of given horse power H, making a given num- 
ber of revolutions -JN per minute, shall cause its angular 

velocity not to deviate more than — th from the mean ; or 

conversely, the mean radius being given, the section K may 
be determined according to these conditions. 



269. In the above equations, m is taken to represent the 
number of effective strokes made by the piston of the engine 
whilst the fly-wheel makes one revolution, and 'i to represent 

that angle whose sine is — . 

Let now the axis of the fly-wheel be supposed to be a 
continuation of the axis of the crank, so that both turn with 
the same angular velocity, as is usually the case ; and let its 
application to the single-acting engine, the double-acting 
engine, and to the double crank engine, be considered sepa- 
rately. 

1. In the single-acting engine, but one effective stroke of 
the piston is made whilst the fly-wheel makes each revolution. 

In this case, therefore, m=l, and sin. »j=— =0*3183098 = 

y, 1 OO QQ' 

sin. 18° 33'; therefore, cos. *j = -9480460, also -= 1QAO = 

•103055; therefore, 1 - — = -793888. 

it 

.-. j ^ cos. ij — 1 1 — ~U = 1-102203. 



THE FLY-WHEEL. 361 

Substituting in equations (345) and (346), 

Hn 
W=95330'64^, 

R^^f.K.755^ (3,7); 

by which equations are determined, according to the pro- 
posed conditions, the weight W in tons of a 11 y- wheel for a 
single-acting engine, its mean radius in feet B, being given, 
and its material being any whatever; and also its mean 
radius R in feet, its section (in square feet) K being given, 
and its material being cast iron of mean quality ; and lastly, 
the section K of its rim in square feet, its mean radius R 
being given, and its material being, as before, cast iron. 

2. In the double-acting engine, two effective strokes are 
made by the piston, whilst the fly-wheel makes one 

2 
revolution. In this cases therefore, m = 2 and sin. r^—-~ 

0-636619 = sin. 39° 32'; therefore, cos. *i = -7712549- = 

39° 39' / 2'i \ 

18Q " =-21963; therefore 1- — J ='56074; 

- cos. n— (l — — ) [ = -21051. 

Substituting in equations (345) and (346), 

Rn 

W= 18207 ^ 2 , 

_, 24-3593 ;/tk Tr „ AnA TIn , ftjm 

by which equations the weight of the fly-wheel in tons, the 
mean radius in feet, and the section of the rim in square: 
feet are determined for the double-acting engine. 

3. In the engine working with two cylinders and a double 
crank, it has been shown (Art. 263.) that the conditions of 
the working of the two arms of a double crank are precisely 
the same as though the aggregate pressure 2P, upon their 
extremities, were applied to the axis of the crank by the 
intervention of a single arm and a single connecting rod; 



THE FRICTION OF THE FLY-WHEEL. 

the length of this arm being represented by — instead of a- 

and its direction equally dividing the inclination of the arms 
of the double crank to one another. 

Now, equations (345) and (34(3) show the proper dimen 
sions of the fly-wheel to be wholly independent of the 
length of the crank arm; whence it follows that the dimen- 
sion of a fly-wheel applicable to the double as well as a 
single crank, are determined by those equations as applied 
to the case of a double-acting engine, the pressure upon 
whose piston rod is represented by 2P,. But in assuming 
il$m . 2P,<z=33000H, we have assumed the pressure upon 
the piston rod to be represented by P, ; to correct this error, 
and to adapt equations (345) and (346) to the case of a 
double crank engine, we must therefore substitute -JH for H 
in those equations. We shall thus obtain 

Jin 

W=9103-5 



N 3 K 2 
K== 



19-3339 i/Hn Tr >. MC% H.n , nAM 



by which equations the dimensions of a fly-wheel necessary 
to give the required steadiness of motion to a double crank 
engine are determined under the proposed conditions. 



The Friction of the Fly-wheel. 

270. W representing the weight of the wheel and <p the 
limiting angle of resistance between the surface of its axis 
and that of its bearings, sin. <p will represent its coefficient 
of friction (Art. 138.), and W sin. 9, the resistance opposed 
to its revolution by friction at the surface of its axis. Now, 
whilst the wheel makes one revolution, this resistance is 
overcome through a space equal to the circumference of the 
axis, and represented by 2*-p, if p be taken to represent the 
radius of the axis. The work expended upon the friction of 
the axis, during each complete revolution of the wheel, is 
therefore represented by 2*pW sin. <p ; and if N represent 
the number of strokes made by the engine per minute, and 

AT 

therefore —the number of revolutions made by the fly-wheel 

2 J 



MODULUS OF THE CKANK AND FLY-WHEEL. 363 

per minute, then will the number of units of work expended 
per minute, upon the friction of the axis be represented by 
KtfpW sin. 9 ; and the number of horses' power, or the frac- 
tional part of a horse's power thus expended, by 

N^PBfa-9 (350). 

33000 v ; 

If in this equation we substitute for "W* the weight in lbs. 
of the fly-wheel necessary to establish a given degree of 
steadiness in the engine, as determined by equations (347), 
(348), and (349), we shall obtain for the horse power lost by 
friction of the fly-wheel, in the single-acting engine, the 
double-acting engine, and the double crank engine, respec- 
tively, the formulsB 

20329^^, 

3882 . 5 1^, 1MM5 Sw (351) 



The Modulus of the Ckanx and Fly-wheel. 

271. If Sj represent the space traversed by the piston of 
the engine in any given time, and a the radius of the crank, 
W the weight of the fly-wheel in lbs., and p the radius of its 

axis, then will 2a represent the length of each stroke, —J the 

2ia 

c 

number of strokes made in that time, and 2tfpW sin. <p . _ i 

2a 

or tfWSj -? sin. <p the work expended upon the friction of the 

fly-wheel during that time, which expression being added to 
the equation (329) representing the work necessary to cause 
the crank to yield a given amount of work U, to the ma- 
chine driven by it (independently of the work expended on 
the friction of the fly-wheel), will give the whole amount of 
work which must be done upon the combination of the crank 
and fly-wheel, to cause this given amount of work to bo 
yielded by it, on the machine which the crank drives. Let 
this amount of work be represented by U„ then in the case 
in which the directions of the driving pressure and the re- 
sistance upon the crank are parallel (equation (829), and the 



364 



TIIE GOVERNOR. 



friction of the crane guide is neglected, we obtain for the 
modulus of the crank and fly-wheel in the double-acting 
engine 



tr,= J i + 



l( ? J sin. o x + 6 sin. ?, V} U, + tWS, I sin. 9 (352). 
2\a a / ) a 



The Governor. 

272. This instrument is represented in the figure, under 
that form in which it is most commonly applied to the steam 
. engine. BD and CE are rods jointed 

at A upon the vertical spindle AF, 
L and at D and E upon the rods DP 

S and EP, which last are again jointed 

at their extremities to a collar fitted 
accurately to the surface of the spin- 
dle and moveable upon it. At their 
extremities B and C, the rods DB 
and EC carry two heavy balls, and 
being swept round by the spindle — 
which receives a rapid rotation al- 
ways proportional to the speed of the 
machine, whose motion the governor 
is intended to regulate — these arms 
by their own centrifugal force, and 
that of the balls, are made to separate, and thereby to cause 
the collar at P to descend upon the spindle, carrying with it, 
by the intervention of the shoulder, the extremity of a lever, 
whose motion controls the access of the moving power to 
the driving point of the machine, closing the throttle valve 
and shutting off the steam from the steam engine, or closing 
the sluice and thus diminishing the supply of water to the 
water-wheel. Let P be taken to represent the pressure of 
the extremity of the lever upon the collar, Q the strain 
thereby produced upon each of the rods DP and EP in the 
direction of its length, TT the weight of each of the balls, w 
the weight of each of the rods BD and CE. AB=#, AD=J, 
DP=<?, FAB=t\ APD=d,. Now upon either of these rods 
as BD, the following pressures are applied : the weight of 
the ball and the weight of the rod acting vertically, the 
centrifugal force of the ball and the centrifugal force of the 
rod acting horizontally, the strain Q of the rod DP, and 
the resistance of the axis A. If a represent the angular 




THE GOVEENOE. 365 

W "W 

velocity of the spindle, — a . FB, or — a asm J, will repre- 

if if 

sent the centrifugal force upon the ball (equation 102), 

W 

and — aV sin. & cos. its moment about the point A ; also 

9 

the centrifugal force of the rod BD produces the same effect 

as though its weight were collected in its centre of gravity 
(Art. 124.), whose distance from A is represented by \{a— h), 
so that the centrifugal force of the rod is represented by 
w 
i—a(a—h) sin. 0, and its moment about the point A by 

if 

w 
^-a(a—hy sin. & cos. &. On the whole, therefore, the sum of 

if 

the moments of the centrifugal forces of the rod and ball are 
represented by — {TW+Jw(«— Vf\ sin. & cos. $. JSTow if m* 

represent the weight of each unit in the length of the rod, 
w = v.{cb+h) ; therefore Wa? + \w{a - Vf = Wa?+^{a? - V) 
(a—b). Let this quantity be represented by W x a\ 

.-. W, = W + J? ( 1 - |) (a-h).... (353) ; 

2 

then will — "W^'sin. & cos. & represent the sum of the moments 

9 
of the centrifugal forces of the rod and ball about A. More- 
over, the sum of the moments of the weights of the rod and 
ball, about the same point, is evidently represented by Wa 
sin. & + wi(a—b) sin. d, or by IWa+^cf—b*)] sin. & ; let this 
quantity be represented by W 2 #sin. d, 

.-. W 2 =W + i^(l - Q . . . . (354). 

Also the moment of Q about A=Q . AEI=Q5 sin. (0 + *,). 
Therefore, by the principle of the equality of moments, ob- 
serving that the centrifugal force of the rod and ball tend to 
communicate motion in an opposite direction from their 
weights and the pressure Q, 

i 

— W^sin. 6 cos. 6=zQb sin. (d + ^^ + W^a sin. d. 



360 



THE GOVERNOR. 



Now P is the resultant of the pressures Q acting i:» the 
directions of the rods PD and PE, and inclined to one 
another at the angle 2d, ; therefore (equation 13), 

P=2Qcos.<J i; 

/. Q sin. (4 + fl,) = JP 8in ^ + ^ =-|P Jsin. d + cos. 6 tan. 6 1 . 
cos. 6 l 

But since the sides b and c of the triangle APD are oppo- 
site to the angles 6 X and d, therefore sin - i _g, • therefore 

sin. d c 

cos. *,= (l-A'sin. 2 *) V; 

.\Qsin. (d + ^) = -|P jsin. d + -sin. d cos. 6 ( 1 — -ysin.'d 1 |. 

Substituting this value in the preceding equation, dividing 
by sin. d, and writing (1— cos. *6) for sin. a d, we obtain 

!^W 1 a 1 cosJ = -Po \ 1 + ^cos.^l-^-f 
? 2 ( e \ & 

^cos.^)~* I + W,a...(355); 

which equation, of four dimensions in terms of cos. 0, being 
solved in respect to that variable, determines the inclination 
of the arms under a given angular velocity of the spindle. 
It is, however, more commonly the case that the inclination 
of the arms is given, and that the lengths of the arms, 
or the weights of the balls, are required to be determined, 
so that this inclination may, under the proposed conditions, 
be attained. In this case the values of "W, and W a must be 
substituted in the above equation from equations (353) and 
(354), and that equation solved in respect to a or W. 

The values of b and c are determined by the position on 
the spindle, to which it is proposed to make the collar 
descend, at the given inclination of the arms or value of d. 
If the distance AP, of this position of the collar from A, be 
represented by h, we have h=b cos. b + c cos. #„ 

.\il=6co6.d + <?(l - £ sin.'c) V (356); 



THE GOVERNOR. 367 

from which equation and the preceding, the value of one 
of the quantities b or c may be determined, according to the 
proposed conditions, the value of the other being assumed to 
be any whatever. 

If jN" represent the number of revolutions, or parts of a 
revolution, made per second by the fly-wheel, and y'N the 
number of revolutions made in the same time by the spindle 
of the governor, then will 2'ty'N represent the space a de- 
scribed per second by a point, situated at distance unity from 
the axis of the spindle. Substituting this value for a in 
equation (355), and assuming 5=c, we obtain 

— L — W x ^cos. d=Tb+W,a (357) : 

also by equation (356), 

h=2bcos.& (358). 

Eliminating cos. 6 between these equations, and solving in 
respect to A, 

hg(Pb+Wj) 1 
2*ya*W l * JSP ^ > ' 

Let P (1+^) and P (1—^) represent the values of P 
corresponding to the two states bordering upon motion 
(Art. 140) and let N (1+ 4) and ^ (1— \) be the correspond- 
ing values of 1ST ; so that the variation either way of ^th from 
the mean number !N" of revolutions, may be upon the point 
of causing the valve to move. If these values be respectively 
substituted for P and N in the above formula, it is evident 
that the corresponding values of A will be equal. Equating 
those values of h and reducing, we obtain 



m \ n) 






By which equation there is established that relation between 
the quantities W 2 , a, P, m which must obtain, in order that a 
variation of the number of revolutions, ever so little greater 



363 



THE CARRIAGE-WHEEL. 



than the -th part, may cause the valve to move. Neglect 
ing £ as small when compared with n. 



n=2 , m 



W„^ 



( i+ ¥)= 



which expression, representing that fractional variation in the 
number of revolutions which is sufficient to give motion to 
the valve, is the true measure of the sensibility of the 
governor. 



273. The joints E and D are sometimes 
fixed upon the arms AB and AC as in the 
accompanying figure, instead of upon the 
prolongations of those arms as in the pre- 
ceding figure. All the formulas of the 
last Article evidently adapt themselves 
to this case, if b be assumed =0 (in equa- 
tions 353, 354). The centrifugal force of 
the rods EP and DP is neglected in this 
computation. 




The Carriage-wheel. 

274. "Whatever be the nature of the resistance opposed to 
the motion of a carriage-wheel, it is evidently equivalent to 
that of an obstacle, real or imaginary, which the wheel may 
be supposed, at every instant, to be in the act of surmount- 
ing. Indeed it is certain, that, however yielding may be the 
material of the road, yet by reason of its comjpression before 
the wheel, such an immoveable obstacle, of exceedingly small 
height, is continually in the act of being presented to it. 



275. The two-wheeled carriage. 

Let AB represent one of the wheels of a two- wheeled 
carriage, EF an inclined plane, which it is in the act of as- 
cending, O a solid elevation of the surface of the plane, or an 
obstacle which it is at any instant in the act of passing over, 



THE CARRIAGE-WHEEL. 



869 



P the corresponding trac- 
tion, "W" the weight of the 
wheel and of the load which 
it supports. 

Now the surface of the box 
of the wheel being in the 
state bordering upon motion 
on the surface of the axle, 
the direction of the resist- 
ance of the one upon the 
other is inclined at the limit- 
ing angle of resistance, to a 
radius of the axle at their 
point of contact (Art. 141.). 
This resistance has, more- 
over, its direction through 
the point of contact O of 
the tire of the wheel with the obstacle on which it is in the 
act of turning. If, therefore, OH be drawn intersecting the 
circumference of the axis in a point <?, such that the angle 
GcR may equal the limiting angle of resistance <p, then will 
its direction be that of the resistance of the obstacle upon, 
the wheel. 

Draw the vertical GH representing the weight "W, and 
through H draw HK parallel to OR, then will this line 
represent (to the same scale) the resistance R, and GK the 
traction P (Art. 14.) ; 

. P__GK sin. GHK sin. GHK 




GH~sin. GKII-sin. (PGH- 
sin. WsO 
sin. (PLW-WsO)* 



GHK) : 



Let R=radius of wheel, p=radius of axle, ACO=rj, ACW 
~i=inclination of the road to the horizon, ^inclination of 
direction of the traction to the road. Now WsO=WCO + 

COs, but WCO=i + *fc and E5L^ = <^L Let CO* be re- 
sin. Cell CO 

presented by a, then WsO=i-\-w-{-a,, and 



P • 

sin. a=^ysm. <p 



. . . (360). 



Also PLW=- + < + 0; therefore TIM -WsO =--(?i + *-&); 

24 



370 THE CARRIAGE-WHEEL. 

w 8 in (,+,+») 

COS. (*J+a — 6) X n 

when the direction of traction is parallel to the road, 0=0, 
.\P=W{8in. i+ cos. i tan. (n+a)\ . . . . (362). 

If the road and the direction of traction be both horizontal 
6=i=0, and 

P=Wtan. (u + a) (363). 

In all cases of traction with wheels of the common dimen- 
sions upon ordinary roads, ACQ or y\ is an exceedingly small 
angle ; a is also, in all cases, an exceedingly small angle 
(equation 360); therefore tan. (77+ a) =77 + 01 very nearly. 
Now if A be taken to represent the arc AO, whose length 
is determined by the height of the obstacle and the radius 
of the wheel, then 

*=g (364). 

Substituting the value of a from equation (360), 
P=W. < A + f R 8in - 9 > (365). 

276. It remains to determine the value of the arc A inter- 
cepted between the lowest point to which the wheel sinks in 
the road, and the summit O of the obstacle, which it is at 
every instant surmounting. Now, the experiments of Cou- 
lomb, and the more recent experiments of M. Morin,* ap- 
pear to have fully established the fact, that, on horizontal 
roads of uniform quality and material, the traction P, when 
its direction is horizontal, varies directly as the load "W, and 
inversely as the radius R of the wheel; whence it follows 
(equation 365), that the arc A is constant, or that it is the 
same for the same quality of road, whatever may be the 
weight of the load, or the dimensions of the wheel. f The 

* Experiences sur le Tirage des Voitures, faites en 1837 et 1838. (See Ap- 
pendix.) 

f In explanation of this fact let it be observed, that although the wheel 
sinks deeper beneath the surface of the road as the material is softer, yet the 
obstacle yields, for the same reason, more under the pressure of the wheel, the 
arc A being by the one cause increased, and by the other diminished. Also, 
that although by increasing the diameter of the wheel the arc A would be ren- 
dered greater if the wheel sank to the same depth as before, yet that it does 
not sink to the same depth by reason of the corresponding increase of the sur- 
face which sustains the pressure. 



THE CARRIAGE-WHEEL. 371 

constant A may therefore be taken as a measure of the re- 
sisting quality of the road, and may be called the modulus 
of its resistance. 

The mean value of this modulus being determined in re- 
spect to a road, whose surface is of any given quality, the 
value of v will be known from equation (364), and the rela- 
tion between the traction and the load upon that road, under 
all circumstances ; it being observed, that, since the arc A 
is the same on a horizontal road, whatever be the load, if the 
traction be parallel, it is also the same under the same cir- 
cumstances upon a sloping road ; the effect of the slope be- 
ing equivalent to a variation of the load. The same substi- 
tution may therefore be made for tan. (>)+a) in equation 
(362), as was made in equation (363), 

.•.P=W ) sin. l+ ( A + p R sin - * ) cos. « } .... (366). 



277. The lest direction of traction in the two-wheeled 
carriage. 

This best direction of traction is evidently that which gives 
to the denominator of equation (361) its greatest value ; it 
is therefore determined by the equation 

A + p sin. <p . M . 

6=ri-\-a= p .... yob i). 



278. The four-wheeled carriage. 

Let W„ W 2 represent the loads borne by the fore and 
hind wheels, together with their own weights, R„ R 2 their 
radii, p„ p 2 the radii of their axles, and <p„ <p 3 the limiting an- 
gles of resistance. Suppose the direction of the traction P 
parallel to the road, then, since this traction equals the sums 
of the tractions upon the fore and hind wheels respectively, 
we have by equation (366) 

P=W,)sin. l + (A + ^ sin -^ cos. t [ + 
W,jsin.. + (A+P> n -^cos.<), 



372 THE CAERIAGE-WHEEL. 



or, 

P=(W 1 + W,)sin. . + a(^+J?)cob.i + 

| ¥, (£ ) sin. 9.4-W,^ ) sin. ? 2 | cos. «... (368). 

279. The work accumulated in the carriage-wheel.* 

Let I represent the moment of inertia of the wheel about 
its axis and M its volume; then will MR 2 + I represent its 
moment of inertia (Art. 79.) about the point in its circum- 
ferences about which it is, at every instant of its motion, in 
the act of turning. If, therefore, a represent its angular 
velocity about this point at any instant, U the work at that 
instant accumulated in it, and fx the weight of each cubical 

unit of its mass, then (Art. 75.), JJ=^-{M.W+ I) = £-M 

(aRy+ia-I. Now if Y represent the velocity of the axis 
of the wheel, aR=Y; « 

.•.u=^MV°+£t<A; 

whence it follows, that the whole work accumulated in the 
rolling wheel is equal to the sum obtained by adding the 
work which would have been accumulated in it if it had 
moved with its motion of translation only, to that which 
would have been accumulated in it if it had moved with its 
motion of rotation only. If we represent the radius of gyra- 
tion (Art. 80.) by K, I=MK a ; whence substituting and 
reducing, 

U=^m(i+|!)v' (369). 

The accumulated work is therefore the same as though the 
wheel had moved with a motion of translation only, but with 

a greater velocity, represented by the expression 1 1 + -r*A V. 



* For a further discussion of the conditions of the rolling of a wheel, see a 
paper in the Appendix on the Rolling Motion of a Cylinder. 

\ The angular velocity of the wheel would evidently be a, if its centre were 
fixed, and its circumference made to revolve with the same velocity as now. 



ACCELERATED OR RETARDED MOTION. 373 



280. On the state of the accelerated or the retarded 
motion of a machine. 

Let the work U, done upon the driving point of a machine 
be conceived to be in excess of that U 2 yielded upon the 
working points of the machine and that expended upon its 
prejudicial resistances. Then we have by equation (117) 

where V represents the velocity of the driving point of the 
machine after the work U, has been done upon it, V, that 
when it began to be done, and 2^X 2 the coefficient of equable 
motion. Now let S x represent the space through which U, 
is done, and S 2 that through which TJ 2 is done ; and let the 
above equation be differentiated in respect to S 15 

dJJ t .dU 2 dS, _ l Tr dV 

dS l dS 2 dS, g dS t 

/7TT 

but -W- = P x (Art. 51.) if P x represent the driving pressure. 

Also -j^ 1 = P 2 , if P a represent the working pressure ; also 



dS 



^ r dV ^ T dV dt XT dY 1 dV . , . _ 

y m;= y w •ds= v -dt'T=-di=f H uation **)• 

If, therefore, we represent by A the relation-^ 1 , between the 

spaces described in the same exceedingly small time by the 
driving and working points, we have 

P 1 =AAP a +B + -2wX a (370); 

■••/=» • *=£&=* <*«>; 

where/ (Art. 95.) represents the additional velocity actually 
acquired per second by the driving point of the machine, if 
P, and P 2 be constant quantities, or, if not, the additional 
velocity which would be acquired in any given second, if 
these pressures retained, throughout that second, the valuea 
which they had at its commencement. 



374 TUE ACCELERATION OK RETARDATION 



2S1. To deter/nine the coefficient of equable /notion. 

2i0A 2 represents the sum of the weights of all the moving 

elements of the machine, each being multiplied by the ratio 
of its velocity to that of the driving point, which sum has 
been called (Art. 151.) the coefficient of equable motion. It' 
the motion of each element of the machine takes place about 
a fixed axis, and a„ o„ a„ &c., represent the perpendiculars 
from their several axes upon the directions in which they 
receive the driving pressures of the elements which precede 
them in the series, and b v b n . b„ &c., the similar perpen- 
diculars upon the tangents to their common surfaces at the 
points where they drive those that follow them ; then, 
while the first driving point describes the small space AS,, 
the point of contact of the j?x\\ and ^? + lth elements of the 
series will be made (Art. 234.) to describe a space repre- 
sented by 

* A • • • *» 4S,; 
a t a a . . . a p l9 

so that the angular velocity of the ^>th element will be 
represented by 

bA • • • ^ s„ 

a x a^ . . . Op " 

and the space described by a particle situated at distance ,: 
from the axis of that element by 

a t .a 2 . . . a p r ■ 

and the ratio X of this space to that described by the driving 
point of the machine will be represented by 



\ a, . a a . . . a p J r 



The sum l^X 2 will therefore be represented in respect to 
this one element by 



\a 1 .a r ...a p l 



Or if \ p represent the moment of inertia of the element, and 
y-p the weight of each cubic unit of its mass, that portion of 
the value of 2^X 3 which depends upon this element will be 
represented by 



OF THE MOTION OF A MACHINE. 375 

( » A • • • M W 

\ a,a 2 . . . a p I 

And the same being true of every other element of the 
machine, we have 

which is a general expression for the coefficient of equable 
motion in the case supposed. The value of A in equation 
(371) is evidently represented by 

A= W ' ■•■■ h . .... (373). 
a x a^a % . . . a p v J 



282. To determine the pressure upon the point of contact of 
any two elements of a machine moving with an accelerated 
or retarded motion. 

Let^ be taken to represent the resistance upon the point 
of contact of the first clement with the second, p^ that upon 
the point of contact of the second element of the machine 
with the third, and so on. Then by equation (370), observ- 
ing that, Pj and p x representing pressures applied to the 
same element, 2wX 2 is to be taken in this case only in 
respect to that element, so that it is represented by f*^, 

whilst A is in this case represented by — , we have, neglect- 
ing friction, 

p--> i+ 4 1 i 1 . 

af x g 

Substituting the value of f from equation (371), and solving 
in respect to^> x , 

where the value of A is determined by equation (373), and 
that of 2wX a by equation (372). Proceeding similarly in 
respect to the second element, and observing that the 
impressed pressures upon that element are p l and p„ we 
have 



376 ACCELERATED OB RETARDED MOTION. 

f x representing the additional velocity per second of the 
point of application of jp xy which evidently equals — f. 

Qi. 

Substituting, therefore, the value of f from equation (371) 
as before, 

5 a h x P-AP, T 

Substituting the value of p x from equation (374), and solv- 
ing in respect to j? 2 , we have 

*>>= TR P -Tffi i "* + kl "* \ (t5t) • ' * • ( 375 >- 

And proceeding similarly in respect to the other points of 
contact, the pressure upon each may be determined. It is 
evident, that by assuming values of A and B in equations 
(370) and (371) to represent the coefficients of the moduli in 
respect to the several elements of the machine, and to the 
whole machine, the influence of friction might, by similar 
steps, have been included in the result. 



PART IV. 



THEORY OF THE STABILITY OF STRUCTURES. 



General Conditions of the Stability of a Structure of 
Uncemeted Stones.* 

A Structure may yield, under the pressures to which it is 
subjected, either by the slipping of certain of its surfaces of 
contact upon one another, or by their turning over upon the 
edges of one another ; and these two conditions involve the 
whole question of its stability. 



The Line of Resistance. 



283. Let a structure MEXK, composed of a single row of 
uncemented stones of any forms, 
and placed under any given circum- 
stances of pressure, be conceived to 
be intersected by any geometrical 
surface 1 2, and let the resultant a A 
of all the pressures which act upon 
one of the parts MN21, into which 
this intersecting surface divides the 
structure, be imagined to be taken. 
Conceive, then, this intersecting 
surface to change its form and posi- 
tion so as to coincide in succession 
with all the common surfaces of 
contact 3 4, 5 6, 7 8, 9 10, of the 
stones which compose the structure : 
and let &B, cC, dD, eE be the re- 




* Extracted from a memoir on the Theory of the Arch by the author of this 
work in the first volume of the " Theoretical and Practical Treatise on Bridges," 
by Professor Hosking and Mr. Hann of King's College, published by Mr. Wealfl. 
These general conditions of the equilibrium of a system of bodies in contact 
were first discussed by the author in the fifth and sixth volumes of the "Cam- 
bridge Philosophical Transactions." 

S77 



378 THE LINE OF RESISTANCE. 

sultants, similarly taken with a A, which correspond to these 
Beveral planes or intersection. 

In each snch position of the intersecting surface, the result- 
ant spoken of having its direction produced, will intersect 
that surface either wit It hi the mass of the structure, or, when 
that surface is imagined to be produced, without it. If it 
intersect it without the mass of the structure, then the whole 
pressure upon one of the parts, acting in the direction of 
this resultant, will cause that part to turn over upon the 
edge of its common surface of contact with the other part ; 
if it intersect it within the mass of the structure, it will not. 

Thus, for instance, if the direction of the resultant of the 
forces acting upon the part NM 1 2 had been a'A', not inter- 
secting the surface of contact 1 2 within the mass of the 
structure, but when imagined to be produced beyond it to a' ; 
then the whole pressure upon this part acting in a' A.' would 
have caused it to turn upon the edge 2 of the surface of con- 
tact 1 2 ; and similarly if the resultant had been in a" A", 
then it would have caused the mass to revolve upon the 
edge 1. The resultant having the direction #A, the mass 
will not be made to revolve on either edge of the surface of 
contact 1 2. 

Thus the condition that no two parts of the mass should be 
made, by the insistent pressures, to turn over upon the edge 
of their common surface of contact, is involved in this other, 
that the direction of the resultant, taken in respect to every 
position of the intersecting surface, shall intersect that sur- 
face actually within the mass of the structure. 

If the intersecting surface be imagined to take up an infi- 
nite number of different positions, 1 2, 3 4, 5 6, &c, and the 
intersections with it, a, 5, <?, d, &c, of the directions of all 
the corresponding resultants be found, then the curved line 
abechf, joining these points of intersection, may with pro- 
priety be called the line of resistance, the resisting points 
of the resultant pressures upon the contiguous surfaces lying 
all in that line. 

This line can be completely determined by the methods of 
analysis, in respect to a structure of any given geometrical 
form, having its parts in contact bv surfaces also of given 
geometrical forms. And, conversely, the form of this line 
being assumed, and the direction which it shall have through 
any proposed structure, the geometrical form of that struc- 
ture may be determined, subject to these conditions ; or 
lastly, certain conditions being assumed, both as it regards 
the form of the structure and its line of resistance, all that is 



THE LINE OF PRESSURE. 



379 



necessary to the existence of these assumed conditions may 
be found. Let the structure AECD have for its line of re- 
sistance the line PQ. Now 
* it is clear that if this line 
cut the surface MN of any 
section of the mass in a point 
n without the surface of the 
mass, then the resultant of 
the pressures upon the mass 
CMN will act through n, 
and cause this portion of the 
mass to revolve about the 
nearest point N of the in- 
tersection of the surface of 
section MN with the surface of the structure. 

Thus, then, it is a condition of the equilibrium that the 
line of resistance shall intersect the common surface of con- 
tact of each two contiguous portions of the structure actually 
within the mass of the structure ; or, in other words, that it 
shall actually go through each joint of the structure, avoid- 
ing none : this condition being necessary, that no two por- 
tions of the structure may revolve on the edges of their 
common surface of contact. 




. The Line of Pressure. 

284. But besides the condition that no two parts of the 
structure should turn upon the edges of their common sur- 
faces of contact, which condition is involved in the determi- 
nation of the line of resistance, there is a second condition 
necessary to the stability of the structure. Its surfaces of 
contact must no where slip upon one another. That this 
condition may obtain, the resultant corresponding to each 
surface of contact must have its direction within certain 
limits. These limits are defined by the surface of a right 
cone (Art. 139.), having the normal to the common surface 
of contact x at the above-mentioned point of intersection of 
the resultant) for its axis, and having for its vertical angle 
twice that whose tangent is the co-efficient of friction of tlio 
surfaces. If the direction of the resultant be within this 
cone, the surfaces of contact will not slip upon one another; 
if it be without it, they will. 

Thus, then, the directions of the consecutive resultants in 




380 THE STABILITY OF A SOLID BODY. 

respect to the normal to the point, where each intersects ite 
corresponding surface of contact, are to be considered a6 im- 
portant elements of the theory. 

Let then a line ABCDE be taken, which is the locus of 
the consecutive intersections of the 
resultants aA, &B, cC, dD, &c. The 
direction of the resultant pressure 
upon every section is a tangent to 
this line ; it may therefore with pro- 
priety be called the line of pressure. 
Its geometrical form may be deter- 
mined under the same circumstances 
as that of the line of resistance. A 
straight line cC drawn from the point 
Cj where the line of resistance abed 
intersects any joint 5 6 of the struc- 
ture, so as to touch the line of pres- 
sure ABCD, will determine the 
direction of the resultant pressure 
upon that joint: if it lie within the cone spoken of, the 
structure will not slip upon that joint ; if it lie without it, 
it will. 

Tims the whole theory of the equilibrium of any structure 
is involved in the determination with respect to that struc- 
ture of these two lines — the line of resistance, and the line 
of pressure : owe of these lines, the line of resistance, de- 
termining the point of appli cation of the resultant of the 
pressures upon each of the surfaces of contact of the system ; 
and the other, the line of pressure, the direction of that 
resultant. 

The determination of both, under their most general forms, 
lies within the resources of analysis ; and general equations 
for their determination in that case, in which all the surfaces 
of contact, or joints, are planes — the only case which offers 
itself as a practical case — have been given by the author of 
this work in the sixth volume of the " Cambridge Philo- 
sophical Transactions." 



The Stability of a Solld Body. 

285. The stability of a solid body may be considered to be 
greater or less, as a greater or less amount of work must be 
Lone upon it to overthrow it ; or according as the amount 




THE STABILITY OF A STRUCTURE. 381 

of work which must be done upon it to bring it into 
that position in which it will fall over of its own accord is 
greater or less. Thus the stability of the solid represented 
in fiq. 1. resting on a horizontal _,. .. „. n 

t ^ v . D -, -, . Fig. 1. Fig. 2. 

plane is greater or less, according 
as the work which must be done 
upon it, to bring it into the position 
represented in^g. 2., where its cen- 
tre of gravity is in the vertical 
passing through its point of sup- 
port, is greater or less. Now this 
work is equal (Art 60.) to that 
which would be necessary to raise its whole weight, verti- 
cally, through that height by which its centre of gravity 
is raised, in passing from the one position into the other. 
Whence it follows that the stability of a solid body resting 
upon a plane is greater or less, as the product of its weight 
by the vertical height through which its centre of gravity is 
raised, when the body is brought into a position in which it 
will fall over of its own accord, is greater or less. 

If the base of the body be a plane, and if the vertical 
height of its centre of gravity when it rests upon a horizontal 
plane be represented by A, and the distance of the point or 
the edge, upon which it is to be overthrown, from the point 
where its base is intersected by the vertical through its 
centre of gravity, by k ; then is the height through which its 
centre of gravity is raised, when the body is brought into a 
position in which it will fall over, evidently represented by 
(A 3 + & 2 )* — h ; so that if W represent its weight, and U the 
work necessary to overthrow it, then 

TJ=W {(h'+Ff-h} .... (376). 

U is a true measure of the stability of the body. 



The STABiLrrY of a Structure. 

286. It is evident that the degree of the stability of a 
structure, composed of any number of separate but contigu- 
ous solid bodies, depends upon the less or greater degree of 
approach which the line of resistance makes to the extrados 
or external face of the structure ; for the structure cannot be 
thrown over until the line of resistance is so defected as to 






THi: WALL OR PILR. 



intersect the • ~ its direction from 

that surface, when tree from ■ ordinary pressure, the 

leas is 1 »re the probability thai an y such pressure will 

throw it. Tiie nearest distance to which the line ol 
approaches tfa - irill, in the following _ -. 

. and will be called the Moi : a 
Stability of the structure. 

This shortest distance presents itself in the wall and but- 
commonly at the lowest section of the structure. It is 
evidently beneath that point where the line of resistance in- 
■ - rs the lowest section of the structure that the grc 

astance of the foundation should be < . I£ that point 

be firmly supported, no settlement of the structure caL take 
place under the influence of the pressures to which it is ordi- 
narily subject* 



The TTall or Pler. 
287. The stability of a watt. 

If the pressure upon a wall be uniformly distributed along 
its length. 4- and if we conceive it to be intersected by verti- 
cal planes, equidistant from one another and perpendicular 
to its face, dividing it into separate portions, then are the 
conditioi.- ta -'ability, in resr>ect to the pressures applied 
to its entire length, manifestly the same with the conditions 
the stability of each of the individual portions into which it 
is thus divided, in respect to the p. ae - sustained by that 
portion of the wall ; so that if every such columnar portion 
or pier into which the wall is thus divided be construct- 
as 1 -tand under its insistent pressures with any degree of 
firmness or stability, then will the whole structure stand with 
the like deg I firmness or stability: and conversely. 

In the following discussion these equal divis is f the 

length of a wall or pier will be conceived to be made one 

ipart; 90 that in every case the question investigated 

will be that of the stability ^of a column of uniform or varia- 

* A practical rule of Tauban, generally adopted in fortifications, brings the 

point where the fine of resistance intersects the base of the wall, to a distance 

from the vertical to its centre of gravity, of £ ths the distance from the latter 

to the external edge of the base. ^See Poncelet, Memoire tur la Stabiliti de* 

tens, note. | B 

t In the wall of a building the pressure of the rafters of the roof is thui 
uniformly distributed by the intervention of the wall y 



THE LINE OF RESISTANCE IN A PIER. 



383 



ble thickness, whose width measured in the direction of the 
length of the wall is one foot. 



288. When a wall is supported by buttresses placed at 
equal distances apart, the conditions of the stability will be 
made to resolve themselves into those of a continuous wall, 
if we conceive each buttress to be ex- 
tended laterally until it meets the adja- 
cent buttress, its material at the same 
time so diminishing its specific gravity 
that its weight when thus spread along 
the face of the wall may remain the 
same as before. There will thus be ob- 
tained a compound wall whose external 
and internal portions are of different 
specific gravities ; the conditions of 
whose equilibrium remain manifestly 
unchanged by the hypothesis which has 
been made in respect to it. 




yptitafca 



The Line of Resistance ln a Pier. 

289. Let ABEF be taken to repre- 
sent a column of uniform dimensions. 
Let PS be the direction of any pres- 
& sure P sustained by it, intersecting its 
axis in O. Draw any horizontal sec- 
tion IK, and take ON to represent 
the weight of the portion AKIB of 
the column, and OS on the same scale 
to represent the pressure P, and com- 
plete the parallelogram ONRS ; then 
will OR evidently represent, in mag- 
nitude and direction, the resultant of 
the pressures upon the portion AKIB 
of the masa (Art. 3.), and its point of 
intersection Q with IK will represent 
a point in the line of resistance. 
Let PS intersect BA (produced if necessary) in G, and let 
GC=&, AB=a, AK=tf, MQ=v, POO=a, f/.=weight of 
each cubic foot of the material of the mass. Draw EEL per- 
pendicular to CD ; then, by similar triangles, 




384: THE LINE OF RESISTANCE LN A PEER. 

QM_RL 

OM~OL 

But QM=y, OM=CM-CO=z-& cot. _«, RL=RN 
sin. RNL=P sin. a, OL=ON+NL=ON + RN cos. RNL 
=tiax-\-P cos. a ; 

. y P sin. a 

" x—k cot. a - fAaaJ + P COS. a ' 

-p. # sin. a — & COS. a /«m« 

.'.y=P. -p (377); 

^ax-\-r cos. a ' 7 

which is the general equation of the line of resistance of a 
pier or wall. 



290. The conditions necessary that the stones of the pier may 
not slip on one another. 

Since in the construction of the parallelogram ONES, 
whose diagonal OE determines the direction of the resultant 
pressure upon any section IK, the side OS, representing the 
pressure P in magnitude and direction, remains always the 
same, whatever may be the position of IK ; whilst the side 
ON, representing the weight of AKLB, increases as IK de- 
scends : the angle ROM continually diminishes as IK de- 
scends. Now, this angle is evidently equal to that made by 
OR with the perpendicular to IK at Q ; if, therefore, this 
angle be less than the limiting angle of resistance in the 
highest position of IK, then will it be less in every subjacent 
position. But in the highest position of IK, QN=0, so that 
in this position ROM=a. Now, so long as the inclination 
of OR to the perpendicular to IK is less than the limiting 
angle of resistance, the two portions of the pier separated by 
that section cannot slip upon one another (Art. 141.). It is 
therefore necessary, and sufficient to the condition that no 
two parts of the structure should slip upon their common 
surface of contact, that the inclination a of P to the vertical 
should be less than the limiting angle of resistance of the 
common surfaces of the stones. All the resultant pressures 
passing through the point O, it is evident that the line of 
pressure (Art. 281.) resolves itself into t\\&t point. 



THE LIXE OF RESISTANCE IN - A PrER. 



291. The greatest height of the pier. 

At the point where the line of resistance intersects the 
external face or extrados of the pier, y=z\a\ if, therefore, H 
rep resents the corresponding value of «, it will manifestly 
represent the greatest height to which the pier can be built, 
60 as to stand under the given insistent pressure P. Substi- 
tuting these values for x and y in equation (377), and solving 
in respect to H, 

jg- _ P(^ + £)C0S.Ct /g^gs 

Psin. a— JfW 

If P sin. a=riu& 2 , IL=infinity ; whence it follows that in 
this case the pier will stand under the given pressure P how- 
ever great may be the height to which it is raised. 



292. The line of resistance is an equilateral hyperbola. 

Multiplying both sides of equation (377) by the demmi- 
nator of the fraction in the second member, 

y(paz-\-~P cos. a)=Paj sin. a— Vh cos. a; 

dividing by pa, transposing, and changing the signs of all the 
terms, 

P COS. a\ P COS. a. 




adding 

Psin. a/ PcOS. a\ / P COS. a\ P COS. a /, P Sin. a\ 

pa \ pa I \ W> )~ pa \ pa }' 

/Psin. a W PcO?.a\ Pc0S.a/ 7 Psill.aV 

• (-75- -y) \ x + —.^r)=^r\ h + —^Y 

T j. r,M x ± ^ i m. P sin. a TTr¥1 P COS. a . , 

Let Cll be taken equal to . IIT= : and let 

pa ' # pa ' 

VQ= yi , TV=x„ 

25 



386 



THE LINE OF RESISTANCE IN A PIER. 



;.y,= \r Q= VM-MQ=CH-MQ= ^~-y i 

m „ PcOS.a. 

aJl =TV r =HV+TH=aj+ — — » 

. ji^3j& ( k + ~^) = a constant quantity. 

This is the equation of a rectangular hyperbola, whose 
asymtote is TX.* The line of resist- 
or ance continually approaches TX 
therefore, but never meets it ; whence 
it follows, that if TX lie (as shown 
in the figure) within the surface of 
the mass, or if CH<CB or 

Psm,a <k, or 2P sin. a<f*a f , then 

pa 
the line of resistance will no where 
cut the extrados, and the structure 
will retain its stability under the in- 
sistent pressure P, however high # it 
may be built ; which agrees with 
the result obtained in the preceding 
article. 






Hi 



1 



k 



S \>f 

1 



\i 



tryr 

i 
I 



E X 



293. The thickness of the pier, so that when raised to a given 
height it may have a given stability. 

Let m be taken to represent the nearest distance to which 
the line of resistance is intended to approach the extrados ot 
the pier, which distance determines the degree of its stability, 
and has been called the modulus of stability (Art. 286.). It 
is evident from the last article that this least distance will 
present itself in the lowest section of the pier. At this 
lowest section, therefore, y=\a-m. Substituting this value 
for y in equation (377), and also the height h of the pier lor 
x, and solving the resulting quadratic equation in respect to 



a, we shall thus obtain 



/Pcos. a \ 

-\-sr~ m ' + 



tfgp-)- + ^ *.-&=)«*. (..<*» 



* Church's Analyt. Geora. 



h 

Art. 161. 



A WALL SUPPORTED BY SHORES. 



387 



294. To vary the point of application of the pressure P, so 
that any required stability may be given to the pier. 

It is evident, that if in equation (377) we substitute \a— m 
9 for y, and h for a?, the modulus of stability m 
may be made to assume any given value for 
a given thickness a of the pier, by assigning 
a corresponding value to k ; that is, by mov- 
ing the point of application G to a certain 
distance from the axis of the pier, deter- 
mined by the value of h in that equation. 
This may be done by various expedients, 
and among others by that shown in the 
figure. Solving equation (377) in respect to 
Jc, we have 

Jc=htm.a-(ia-m) /l+ J^_ j. . . . (380). 

It is necessary to the equilibrium of the pier, under these 
circumstances, that the line of resistance should no where 
intersect its intrados below the point D. 



■ , 1 , 1 


- /■ 



The Stability of a Wall supported by Shores. 

295. Let the weight of the portion of the wall supported 
by each shore or prop, and the 
pressure insistent upon it, be im- 
agined to be collected in a single 
foot of the length of the wall ; the 
conditions of the stability of the 
wall evidently remain unchanged 
by this hypothesis. Let ABCD 
represent one of the columns or 
piers into which the wall will thus 
be divided, EF the corresponding 
shore, P the pressure sustained upon 
the summit of the wall, Q the 
thrust upon the shore EF, 2w its 
weight, x the point where the line 
of resistance intersects the base of 
the wall, Cx=m, C¥=b, FEC^o 1 ; 
and let the same notation be taken in other respects as iu 




3S8 A WALL SUPrOKTED BY SHORES. 

the preceding articles. Then, since a? is a point in the diree» 
tion of the resultant of the resistances by which the base of 
the column is sustained, the sum of the moments about that 
point of the pressure P and half the weight of the shore, 
supposed to be placed at E*, is equal to the sum of the 
moments of the thrust Q, and the weight pah of the column; 
or drawing a?M and xS perpendiculars upon the directions 
of P and Q, 

P. a?M-fw . a,'C=Q . xls+pah . xK. 

Xow a?M = xs sin. a , *M=(IIK— Ht) sin. a = \h—(Rp + st) 
cot. aj sin. a=A sin. a— (k+^a— mj cos. a, xN=(b + ?n) cos. /3, 

/. P \h sin. a — (Jc + $a— m) cos. a\ -f 
wm=Q(b + m) cos. P + i*>ah(ia—m) 

Solving this equation in respect to Q, and reducing, we 
obtain. 

_P j^sin.a— (Z , + -J-o)cos.aj — %pa*h + m(P cos. a + pa h + w) 
^~ (b + m)cos.fif ,g gl v 

This expression may be placed under the form 
Q=(P cos. a + pah + w) sec. fi — 

P{5cos. a— A sin. a + (& + £«) cos. af + pah (ia + b) + wb^ 
(b + m) cos. /3f 

If the numerator of the fraction in the second member of 
this equation be a positive quantity (as in all practical cases 
it will probably be found to be) the value of Q manifestly 
diminishes with that of m. Now the least value of m, con- 
sistent with the stability of the wall, is zero, since the line 
of resistance no where intersects the extrados; the least 
value of Q (the shore being supposed necessary to the sup- 
port of the wall) corresponds, therefore, to the value zero of 
m ; moreover this least value of the thrust upon the shore 
consistent with the stability of the wall is manifestly that 
which it sustains when the wall simply rests upon it, the 

* The weight 2m of the shore may be conceived to be divided into two equal 
part< and collected at its extremities. 

f The expression (/>-\-m) cos. 3 may be placed under the form b cot 3 sin. 
3-\-m cos. 3=c sin. B-\-w cos. J, where c represents the height CE ol" the point 
against which the prop rests. 



A WALL SUPPORTED BY SHORES. 



389 



shore not being driven so as to increase the thrust sustained 
by it beyond that just necessary to support the wall.* 
This least thrust is represented by the formula 

n __ P {A sin, a— (& + !<%) cos. a \ — ^g 2 h 

The thrust which must be given to the prop in order that 
there may be given to the wall any required stability, deter- 
mined by the arbitrary constant ra, is determined by equa- 
tion (381). The stability will diminish as the value of m is 
increased beyond \a, and the wall will be overthrown 
inwards when it exceeds a. 



296. The stability of a wall sustained by more than one 
shore in the same jplane. 

Let EF, ef be shores in the same plane, sustaining the 
wall ABCD, and both necessary to 
its stability; so that if EF were re- 
moved, the wall would turn over upon 
f and if ef were removed, upon some 
point between F and C. 

If the thrust of the shore EF be 
only that just necessary to sustain 
the tendency of the wall to overturn 
upon f it is evident that the line of 
resistance must pass through that 
point ; but if the thrust exceed that 
just necessary to the equilibrium, or 
if the shore be driven then the line 
of resistance will intersect fg in some 
points. Tuetfx=m ; then represent- 
ing the thrust upon EF by Q, the dis- 
tances /D and fi by h and ft, and the angle EFC by £, the 
value of Q is evidently determined by equation (381). 

If z be taken in like manner to represent the point where 
the line of resistance intersects the base of the wall, and 
Cs=m» CE=5 1 ; Ce=b„ Cfe=P» OD=A„ the thrust upon 
the prop <?/by Q, and its weight by 2w x ; then the sum of 
the moments about the point z of Q and Q„ and the weight 




c z 



* This case presents an application of the principle of Uasi resistance. (See 
Theory of the Arch.) 



390 



THE STABILITY OF A GOTHIC STRUCTURE. 



fAaA, of tlie wall, equals the sum of the moments of P, w t 
and w x ; or 

QX^ + m,) cos. ft + Q (Jj+m,) cos. /3-f-fAaA, (%a—m)-= 
P{A X sin. a—(k-\-^a—m 1 ) cos. a} + (w+w t ) m x (3S2.) 

Substituting the value of Q in this equation, from equation 
(381), and solving in respect to Q„ the thrust upon the prop 
cf will be determined, so that the stability of the wall, upon 
its section fg and upon its base CB, may be m and m x 
respectively. 

If m 1 =m, the portions of the wall above and below fg 
are equally stable. 

If m 1 =?n=0, the thrust upon each shore is only that 
which is just necessary to support the wall, or which is pro- 
duced by its actual tendency to overturn. In this case we 
have 



Q,= 



(P sin, a— jiig) (hfi—hb,) + ~? (b.—b) (k+ja) cos, a 



bb, cos. (S l 
the value of k being determined by equation (380). 



297. The stability of a structure having parallel walls, one 
of which is supported by means of struts resting on the 
summit of the other. 

Let AB and CD be taken to represent the walls, and EF 
one of the struts ; the thrust Q upon the 
strut may be determined precisely as in 
Art 295. So that the line of resistance 
may intersect the base of the wall AB at 
a given distance m from the extrados 
(see note, p. 388.) 

Let m, represent the distance Dx from 
the extrados at which the line of resist- 
ance intersects the base of the wall CD ; 
then taking the moments of the pressures 
applied to the wall CD about the point 
i a?, as in Art. 295, and observing that 

Sf besides the pressure Q the weight w of 

one half the strut is applied at E, we 
have 

Q \h t sin. P+&— fa— «0 cos. (3\ =^ 1 a 1 h l (J^— m,) + 
(k 1 + ia 1 —?n i )w y 




THE STABILITY OF A GOTHIC STRUCTUBE. 391 

in which equation \ and a x are taken to represent the 
height and thickness of the wall CD, & x the distance of the 
point E on which the strut rests from the axis of the wall, (3 
the inclination of the strut to the vertical, and ^, the weight 
of a cubic foot of the material of the wall. 

Substituting for Q its value from equation (381), and 
reducing, 

P\hsin.*— (k+ia) cos. a,}— i^efh+m (P cos: a + pah+w)__ 
c sin. {3+m cos. /3 



\ sin. (3 — (& x -f \a x — m x ) cos. /3 



(383). 



By this equation is determined that relation between the 
dimensions of the two walls and the amount of the insistent 
pressure P, by which any required stability may be assigned 
to each wall of the structure. If m=0, the pressure upon 
the strut will be that only which is produced by the ten- 
dency of AB to overturn ; and the value of m x determined 
from the above equation will give the stability of the exter- 
nal wall on this supposition. 

If m=0 and m 1 =0, both walls will be upon the point of 
overturning, and the above equation will express that rela- 
tion between the dimensions of the wall and the amount of 
the insistent pressure, which corresponds to the state of the 
instability of the structure. 

The conditions of the stability, when the wall AB is sup- 
ported by two struts resting upon the summit of the wall 
CD, may be determined by a method similar to the above 
(see Art 296). 

The general conditions of the stability of the structure 
discussed in this article evidently include those of a Gothic 
Building having a central nave, whose walls are supported, 
under the thrust of its roof, by the rafters of the roof of its 
side aisles. By a reference to the principles of the preceding 
article, the discussion may readily be made to include the 
case in which a further support is given to the walls of the 
nave by flying buttresses, which spring from the summits of 
the walls of the aisles. The influence of the buttresses 
which support the walls of the aisles upon the conditions of 
the stability of the structure forms the subject of a subse- 
quent article. 



392 



THE WALL OF A DWELLING. 



298. The stability of a wall sustaining the floors of a 
dwelling. 

The joists of the floors of a dwelling-house rest at their 
extremities upon, and are sometimes 
notched into, pieces of timber called 
wall-plates, which are imbedded in 
the masonry of the wall. They 
serve thus to bind the opposite sides 
of the house together ; and it is upon 
the support which the thin walls of 
modern houses receive from these 
joists, that their stability is some- 
times made to depend.* 

Representing by w the weight of 
that portion of the flooring which 
rests upon the portion ABCD of the 
wall, and the distance BE by c, 
taking x, as before, to represent the 
point where the line of resistance 
intersects the base of the wall, and 
measuring the moments from this 
point, we have 

xN . Q + xK . fiah + xB. w=xK. P ; 

whence, taking the same notation as in the preceding arti- 
cles, and substituting, 

cQ + (ia—m)pah + (a—m)w={hsm.a.—(k+%a--m) cos. a} P ; 

,\ Qc= \h sin. a—(k+ia) cos. a} P— \^h— wa+ 

m (P cos. a+pah+w) (384) ; 

from which expression it appears that Q is less as m is less. 
When, therefore, the strain upon the joints is that only 
which is just necessary to preserve the stability of the wall, 
or which it produces by its tendency to overturn, then 
m=0. In this case, therefore, 




* A house thus constructed evidently becomes unsafe when its wall-platea Of 
the extremities of its joists begin to decay. 



A WALL SUPPORTED BY BUTTRESSES. 393 

{A sin. a— (k+ia)cos. a}P— \\m£K— wa 
y= " " ~ G (385). 

If (3 be assumed a right angle, and if (a—m)w be substi- 
tuted for mw, the case discussed in Art. 295. will 
evidently pass into that which is the subject of 
the present article, and the preceding equation 
may thus be deduced from equation (381) (see 
note, p. 388.). 

In like manner, if the wall sustain the pres- 
sure of two floors, and A be taken to represent 
the distance from its summit to the lower floor, 
and Aj its whole height ; then, representing by m 
and m x the distances from the extrados at which 
the line of resistance intersects the sections EG 
3 and eg, and substituting (w + w^) (a-m,) for 
(w+w^m^ the value of the strain Q on the 
joists of the lower floor may be determined by 
equation (382), it being observed that for the 
coefficient of Qj in that equation must be substi- 
tuted (as was shown above) the height (h 1 —h) of 
the lower floor from the bottom of the wall. If the strain 
be only that produced by the tendency of the wall to over- 
turn at g and C, then 

Q iG= (h—c) Qpa'—'P sin. a)-f- 
Y(Jc+ia) cos. a +wa— h l _ h (386). 




The value of Q is determined by equation (385), c beinoj 
taken to represent the distance E^ between the floors. It 
the joists be not notched into the wall-plates, the friction of 
their extremities upon them, produced per foot of the length 
by the weight which they support, must at least equal Q and 
Q x respectively. 



299. The stability of a wall supported by piers or buttresses 
of uniform thickness. 

Let the piers be imagined to extend along the whole 



394 



WALL SUPPORTED BY BUTTRESSES. 



D G 



§ 



s 



length of the wall, as explained in Art. 288. : 
and let ABCD represent a section of the com- 
pound wall thus produced. Let the weight of 
each cubic foot of the material of the portion 
ABFE be represented by i^, and that of each 
cubic foot of GFCD by f*,, EA=a I? GD = a„ 
BC=a, AB = A n CD = A 2 , distance from CD, 
produced, of the point where P intersects 
AE=Z, x the intersection of the line of resist- 
ance with CB, Qx—m. By the principle of the 
equality of moments, the moment of P about 
the point x is equal to the sum of the moments 
of the weights of GC and AF about that point. 
But (Art. 295.) moment of P=P \h, sin. a — 
(l—m) cos. a\ ; also moment of weight of AF= 
(# 2 — m +%a 1 )h 1 a 1 v< 1 ; moment of weight of GC= 

:. P |Aj sin. a— {l—m) cos. *} =(#„— m+^aj A^^-j- 
(ia 2 -m)A,a^ 2 (387). 

If the material of the pier be the same with that of the 
wall ; then, taking b to represent the breadth of each pier, 
and c the common distance of the piers from centre to 
centre (Art. 288.), ca^=ba^ x , therefore c^ i =b^ 1 . Repre- 

senting y- by n, eliminating the value of m- 3 between this 

equation and equation (387), writing p for f^, and reducing, 

P(A 1 sin. a— I cos. *)=iiJ<la*h 1 + 2a 1 a 2 h 1 + -a*hA -- 

m)F cos.a + v(a 1 h 1 +-a n h] (• • • .(388); 

by which equation a relation is determined between the 
dimensions of a wall supported by piers, having a given 
stability m, and its insistent pressure P. Solving it in 
respect to a 3i the thickness of the pier necessary to give any 
required stability to the wall will be determined. (See 
Appendix.) 

If a 2 be assumed to represent that width of the pier by 
which the wall would just be made to sustain the given 
pressure P without being overthrown; then taking ra=0, 
and solving in respect to a» 



WALL SUPPORTED BY BUTTRESSES. 



395 



a 2 = — na x j- + 



/gift sin. a-l cos. -)+^ (^ -1)) *',... (389). 




300. TAtf stability of a pier or buttress sur* 
mounted by a pinnacle. 

Let W represent the weight of the pinnacle, 
and e the distance of a vertical through its cen- 
tre of gravity from the edge C of the pier : then 
assuming x to be the point where the line of 
resistance intersects the base of the pier, and tak- 
ing the same notation as before, equation (387) 
will evidently become 



P \\ sin. «— (I— m) cos. *] = \a i —m+ia 1 ] h x a^ x -\- 
{ia 2 —m\ h a a^ + (e—m)W. 

Substituting for /* a its value-f^ or—, writing f* for p l9 and 

C 7h 

reducing, 

P^sin. a— Zcos. a)=^( < 2 1 2 A 1 H-2« 1 ^ 2 A 1 H — a*hA 4- 

W<?-m JPcos. cc+W + ^aA + hiA) }- • - (390). 

If <z 2 represent the thickness of that pier by which the wall 
will just be sustained under the pressure, taking m=0, and 

solving in respect to a„ a 2 =—na^-[- 

l/|j {Pfo sin. «-l cos. «)-We\ +^ (^-l) < . . (391). 



396 



WALL SUPPOKTK1) BY GOTHIC BUTTRESSES. 



The Gothic Buttress. 



301. In Gothic buildings the thickness of 
a buttress is not unfrequentlv made to vary 
at two or three different heights above its 
base. Such buttress is represented in the 
accompanying figure. 

The conditions by which any required sta- 
bility may be assigned to that portion of it 
whose base is he may evidently be determined 
by equation (390). To determine the condi- 
tions of the stability of the w T hole buttress 
upon CD, let the heights of the points Q, «, 
and b above CD be represented by A,, A 2 and 
A 3 ; let DE=^, DF=a„ FC=fl„ Cx=m x ; 
then adopting, in other respects, the same 
notations as in Arts. 299 and 300. Since the 
distances from a? of the verticals through the 
centres of gravity of those portions of the 
buttress whose bases are DE, DF, and FC 
respectively, are (& 3 -f- a 2 + ■£-«, — m,), (# 3 + -£-# 2 
«•■■ » a _ 7#J and {^a^—m^ we "have, by the equality 

of moments, 

P jAj sin. a— (I— ra^cos. aj =(a i + a t +ia l —m^h l a l i* + 
fa+&-m, l )h t a*+(fa-m l )h t a?-+yf(e-m l ) (392). 

7b 7b 




This equation establishes a relation between the dimen- 
sions of the buttress and its stability, by which any one of 
those dimensions which enter into it may be so determined 
as to give to m x any required value, and to the structure any 
required degree of stability. (See Appendix.) 

It is evident that, with a view to the greatest economy of 
the material consistent with the given stability of the but- 
tress, the stability of the portion which rests upon the base 
le should equal that of the whole buttress upon CE ; the 
value of ?n 1 in the preceding equation should therefore equal 
that of in in equation (390) If m be eliminated between 
these two equations, it being observed that li x and A a in equa- 
tion (390) are represented by h x —h z and /t 2 —/i 3 in equation 
(392), a relation will be established between <z„ a„ a 3 A„ h„ 
A 3 , which relation is necessary to the greatest economy of 




THE STABILITY OF WALLS SUSTAINING EOOFS. 397 

material ; and therefore to the greatest stability of the struc* 
ture with a given quantity of material. 



The Stability of Walls sustaining Roofs. 

302. Thrust upon the feet of the rafters of a roof the tie- 
beam not being suspended from the ridge. 

If f*j be taken to represent the weight of each square foot 
of the roofing, 2L the span, i the 
inclination BAG of the rafters to 
the horizon, q the distance between 
each two principal rafters, and a 
the inclination to the vertical of 
the resultant pressure P on the 
foot of each rafter ; then will L sec. t represent the length of 
each rafter, and \*>^Lg sec. i the weight of roofing borne by 
each rafter. Let the weights thus borne by each of the 
rafters AB and BC be imagined to be collected in two equal 
weights at its extremities ; the conditions of the equilibrium 
will remain unchanged, and there will be collected at B the 
weight supported by one rafter and represented by f^L^ 
sec. t, and at A and C weights, each of which is represented 
by i^^Lq sec. i. Now, if Q be taken to represent the thrust 
produced in the direction of the length of either of the 
rafters AB and BC, then (Art. 13.) ^Lq sec. t = 2Q cos. 
JABC: but ABC^tf — 2*; therefore cos. |ABC = sin. t; 
therefore 2Q sin. 1=^1^ sec. i ; 

' l -^2 sin. i 2 sin. t cos. t> sin. 2t* 

The pressures applied to the foot A of the rafter are the 
thrust Q and the weight ii^^Lq sec. i ; and the required pres- 
sure P is the resultant of these two pressures. Resolving Q 
vertically and horizontally, we obtain Q sin. i and Q cos. *, 
or i^^Lq sec. i and fy^Juq cosec. t. The whole pressure applied 
vertically at A is therefore represented by pJLq sec. t, and 
the whole horizontal pressure by i'^J^q cosec. i ; whence it 
follows (Art. 11.) that 

P = V^Uq* sec. 2 ^ + i,a l 3 LY cosec. h= 
M^sec. i Vl+icot. s < (393). 



,°>0S RAFTERS OF A EOOF. 

^u Ly C086C. i , , oft N 

tan.a = - / Q „ , =&COt.l (394). 

fAjLy sec. I ■ J 

If the inclination i of the roof be made to vary, the span 
remaining the same, I* will attain a minimum value when 

tan. i = , or when 

t=35° 16' (395). 

It is therefore at this inclination of the roof of a given 
span, whose trusses are of the simple form shown in the 
figure, that the least pressure will be produced upon the feet 
oi the rafters. If? represent the limiting angle of resistance 
between the feet of the rafters and the surface of the tie, the 
feet of the ratters would not slip even if there were no mor- 
tice or notch, provided that a were not greater than <p (Art. 
141.), or £ cot. 1 not greater than tan. 9, or 

1 not less than cot.- 1 (2 tan. <p)* (396). 



303. The thrust upon the feet of the rafters of a roof in 
which the tic-learn is suspended from the ridge by a 
king-post. 

It will be shown in a subsequent portion of this work 
(see equation 558) that, in this case, 
the strain upon the king-post BD is 
equal to f ths of the weight of the 
tie-beam with its load. Kepresent- 
r ing, therefore, the weight of each 
foot in the length of the tie-beam 
by fj. s , and proceeding exactly as in 

the last article, we shall obtain for the pressure P upon the 

feet of the rafters, and its inclination to the vertical, the 

expreseioDfl 

P=jL{(3fi.^sec.*+^ , +(M. 1 2'sec.t+*f*,) , cot. f t}* (397). 

te „t.(M22±fc£) (398). 

* If the surfaces of contact bo oak, and thin slips of oak plank be fixed 
under the feet of the rafters, so that the surfaces of contact may present par- 
ullrl fibres of the wood to one another (by which arrangement the friction will 
itlv increased), tan. q = --1S (see p. 133.); whence it follows that the 
r;itt. n will not slip, provided that their inclination exceed cot." 1 -96, or 
40 10\ 




WALL SUSTAINING THE THRUST OF A ROOF. 



399 



304:. The stability of a wall sustaining the thrust of a roof 
having no tie-beam. 

Let it be observed, that in the equation to the line of 
resistance of a wall (equation 
377), the terms P sin. a and P 
cos. a represent the horizontal 
and vertical pressures on each 
foot of the length of the summit 
of the wall ; and that the former 
of these pressures is represented 
in the case of a roof (Art. 302.) 
by ij^L cosec. i, and the latter 
by (x x L sec. i ; whence, substitu- 
ting these values in equation 
(377), we obtain for the equation 
to the line of resistance in a wall 
sustaining the pressure of a roof, 
without a tie-beam 




>ct-k _ 



I— J oxco8.i+Jj 



in which expression a represents the thickness of the wall, 
k the distance of the feet of the rafters from the centre of 
the summit of the wall, L the span of the roof, m- the weight 
of a cubic foot of the wall, and f*, the weight of each square 
foot of the roofing. Tiie thickness a of the wall, so that, 
being of a given height A, it may sustain the thrust of a 
roof of given dimensions with any given degree of stability,. 
may be determined precisely, as in Art. 293, by substituting 
h for x in the above equation, and \a—m for y, and solving 
the resulting quadratic equation in respect to a. 

If, on the other hand, it be required to determine what 
must be the inclination i of the rafters of the roof, so that 
being of a given span L it may be supported with a given 
degree of stability by walls of a given height h and thick- 
ness a J then the same substitutions being made as before, 
the resulting equation must be solved in respect to t instead 
of a. 

The value of a admits of a minimum in respect to the 
variable i. The value of i, which determines such a mini- 
mum value of a, is that inclination of the rafters which is 



400 • A WALL 

- with t! _ my in the material of the 

g given. 



call suppor **es, and 

The conditions of the stability of such a wall, when sup- 
<\ by bir sees rm thickness, will evidently be 

determined, if in equatk : Bfi - ibstitute far P c 

and P sin. a their value* -«• .L Bee. i and Jf*,L consec. *; we 
6hall thus obtain 

i . — Bee. = i^ ( a ?h.~- ^ — " <z j , ^i)~ m 

-H fc («A+— «A \ *oo> 



. which equation the thickness (7. of the buttresses 
nece— _ any required stabili* he wall may 

be determined. 

I: :he thick: at the buttresses be different at different 
heigh* e surmounted by pinnacles, the con- 

ditio - ibflity are similarly determined by substi- 

tutii:_ -:n. a and P cos. a the same values in equations 

_ 
determine the conditions of the stability of a Gothic 
building, wh - 5, having a roof without a tie-beam, is 

supported by the raft- r by flying bnt- 

liich rest npon the summits of the walls of its 
-imilar substitution must be made in equation (383). 
If the : the ai^ 1 by buttresses, 

33) must be replaced by a similar relation 
ned by the methods laid down in A 2 
.me substitution l^r P sin. a and P cos. a must then be 
ma-. 



306. The conditions of the sta f a wall supporting a 

sh- 

Let AB represent one of the rafters oi such a roof, one ex- 



STABILITY OF A WALL. 



dtOl 




tremit y A resting against the face of 
the wall of a building contiguous to 
the shed, and the other B upon the 
^ summit of the wall of the shed. 

It is evident that when the wall 
BH is upon the point of being over- 
thrown, the extremity A will be upon 
the point of slipping on the face of 
the wall DC ; so that in this state of 
the stability of the wall BH, the direc- 
tion of the resistance R of the wall 
DC on the extremity A of the rafter 
will be inclined to the perpendicular AE to its surface at an 
angle equal to the limiting angle of resistance. Moreover, 
this direction of the resistance R which corresponds to the 
state bordering upon motion is common to every other state ; 
for by the principle of least resistance (see Theory of the 
Arch) of all the pressures which might be supplied by the 
resistance of the wall so as to support the extremity of the 
rafter, its actual resistance is the least. Now this least re- 
sistance is evidently that whose direction is most nearly ver- 
tical ; for the pressure upon the rafter is wholly a vertical 
pressure. But the surface of the wall supplies no resistance 
whose direction is inclined farther from the horizontal line 
AE than AR ; AR is therefore the direction of the resist- 
ance. 

Resolving R vertically and horizontally, it becomes R sin. 
9 and R cos. 9. Representing the span BE by L> the incli- 
nation ABF by t, the distance of the rafters by q, and the 
weight of each square foot of roofing by ^ (Art. 10.), R sin. 
9-f-P cos. a=p 1 Lq sec. 1 and R cos. 9— P sin. a=0 ; also* the 
perpendiculars let fall from A on P and upon the vertical 
through the centre of AB, are represented by 

L cos. (a + j) sec. t and -J-L ; therefore (Art. 7). 
PL cos. (a + i)sec. i=ilj . I^s q sec. t, and hence 
P cos. (a,-\-i)=% 1 Lp 1 q. Eliminating between these equa- 
tions, we obtain 



cot. a = tan. (p + 2 tan. 1 (401); 

sm.(9 + * ri1 cos. t (tan. 9 + tan.. 1) K J 



2fi 



402 



Till'. PLATK BA-NDE. 

If the rafter, instead of resting at A 

Bpe rtnre, u shown in the fcgnre, 
M that tl , ■ resistance of the wall mat be 

; ; ^ u,,nn its intVrior sufaee .nstead ot 

rtita cxlremity: then drawing AK er- 
pendicular to the surface of the ratter, 
Redirection Alt of the resistance » eu- 
„lv inclined to that line at the g.ven 
ffig angle,. Its inclination to the hori- 
zon is therefore represented by g - '* 9 " 

Sul.s.imting this angle for 9 in equations (401) and (402), 
cot.«=cot.(*-9) + 2tan.. . . . 




B= 



^T>,? sec, t p_ 



: ^ iA i-5) + sm.^-?)tan. « ' 

51 + [cot.((-<p) + 2tan.j]'!* 
- I L "i 2 ■ _ cos7Tjcot7(^ :: ?)l- tan.t T 



(404). 



- in equations (377) and (379) for P sin. « P cos. «, 
',1 rvduer determined above, all the .conditions of the sta- 
b litv of a trail supporting such a roof will be determined. 



307. The plate bande or straight arch. 

Let MN represent any joint of 
the plate bande ABCD, whose 
points of support are A and B ; 
PA the direction of the resistance 
at A, TVQ a vertical through the 
centre of gravity of AMND, TR 
the direction of the resultant pres- 
sure upon MN ; the directions of 
TR WQ and PA intersect, therefore, in the same point O. 
I* , OAD=«, AM=s, MK=y, AD=H, AB=2L, weight 
n\- cubic foci Of material of arch=Mv Draw K?n a perpen- 
dicular upon PA produced; then by the principle oi tne 
ality of moiiK-nts, 

E^ . P = MQ . (weight of DM). 




THE PLATE BANDE. 403 

But Em = x cos. a — y sin. a, MQ = %x, weight of DM = 
H^a?; also resolving P vertically, 

Pcos.a=LHf*, (405). 

Whence we obtain, by substitution in the preceding equa- 
tion, and reduction, 

L(a?— 2/tan.a)=-^ a (406), 

which is the equation to the line of resistance, showing it to 
be a parabola. If, in this equation, L be substituted for a?, 
and the corresponding value of y be represented by Y, there 
will be obtained the equation Y tan. a = |L, whence it 
appears that a is less as Y is greater ; but by equation (405), 
P is less as a is less. P, therefore, is less as Y is greater ; 
but Y can never exceed H, since the line of resistance can- 
not intersect the extrados. The least value of P, consistent 
with the stability of the plate bande, is therefore that by 
which Y is made equal to H, and the line of resistance 
made to touch the upper surface of the plate bande in F. 

Now this least value of P is, by the principle of least 
resistance (see Theory of the Arch), the actual value of the 
resistance at A, 

.\tan. a =ijj (407). 

Eliminating a between equations (405) and (407), 



P^LH^j/l+ig . ... . .(408). 

Multiplying equations (405) and (407) together, 

Psima^LX (409). 

Now P sin. a represents the horizontal thrust on the point 
of support A. From this equation it appears, therefore, that 
the horizontal thrust upon the abutments of a straight arch 
is wholly independent of the depth H of the arch, and that 
it varies as the square of the length L of the arch ; so that 
the stability of the abutments of such an arch is not at all 
diminished, but, on the contrary, increased, by increasing 
the depth of the arch. This increase of the stability of the 
abutment being the necessary result of an increase of the 
vertical pressure on the points of support, accompanied by 
no increase of the horizontal thrust upon them. 



404 



Tin: PLATE uande. 



30S. The loaded plate bande. 



1 



It is evident that the effect of a loading, distributed 
uniformly over the extrados of the 
plate bande, upon its stability, is in 
every respect the same as would be 
produced if the load were removed, 



— 



__ 






d 



„. 



i\ 



: 



m 



CC 



O 



and the weight of the material of 



the bande increased so as to leave 
the entire weight of the structure 
unchanged. Let ^ s represent the 
weight of each cubic foot when thus 
increased, fx, the weight of each 

cubic foot of the load, and H x the height of the load ; then 

(SHL=fi 1 HL+fsH x L, 



V 



\fS=f*i + lS- 



H 



(410). 



The conditions of the stability of tlie loaded plate bande 
are determined by the substitution of this value of p % for m- x 
in the preceding article. 



309. Conditions necessary that the voussoirs of a plate bande 
may not slip upon one another. 

It is evident that the inclination of every other resultant 
sure to the perpendicular to the surface of its corres- 
ponding joint, is less than the inclination of the resultant 

pressure or resistance P, to the 
perpendicular to the joint AD. 
If, therefore, the inclination be 
not greater than this limiting an- 
gle of resistance, then will every 
other corresponding inclination 
be less than it, and no voussoir 
will therefore slip upon the sur- 
face of its adjacent voussoir. Now the tangent of the incli- 
nation P to tne perpendicular to AD is represented by cot. a 

or by ~j (equation 407) ; the required condition is therefore 

determined by the inequality, 

211 

-=- < tan. <p . . . 




(411). 



THE SLOPING BUTTRESS. 405 

It is evident that the liability of the arch to failure by the 
slipping of its voussoirs, is less as its depth is less as com- 
pared to its length. In order the more effectually to pro- 
tect the arch against it, the voussoirs are sometimes cut of 
the forms shown by the dotted lines in the preceding figure, 
their joints converging to a point. The pressures upon the 
points A and B are dependent upon the form of that portion 
of the arch which lies between those points, and indepen- 
dent of the forms of the voussoirs which compose it ; these 
pressures, and the conditions of the equilibrium of the piers 
which support the arch, remain therefore unchanged by this 
change in the forms of the voussoirs. 

310. To determine the conditions of the equilibrium of 
the upright piers or columns of masonry which form the 
abutments of a straight arch, supposing them to be termi- 
nated, as shown in the figure, on a different level from the 
extrados CD of the arch, let b be taken to represent the 
elevation of the top of the pier above the point A ; then will 

b tan. a, or \ -^ (equation 407), represent the distance AG 

(p. 383), or the value of Jc— \a). Substituting for h in equa- 
tion (377) and also the values of P sin. a, P cos. a, from 
equations (409) and (405), we have 

x— lb + j-a\ 

y= ^~nr\ — — (412); 

which is the equation to the line of resistance of the pier, a 
representing its thickness, b the height of its summit above 
the springing A of the arch, L the length of the arch, ^ the 
weight of a cubic foot of the material of the arch or abut- 
ment (supposed the same). 

The conditions of the stability may be determined from 
this equation as in the preceding articles. If the arch be 
uniformly loaded, the value of ^ 3 given by equation (410) 
must be substituted for fi. 1# 



311. The centre of gravity of a buttress whose faces 
are inclined at any angle to the vertical. 

Let the width AB of the buttress at its summit be repre- 



106 



THE SLOPING Bl l HLEBS. 




Bented bj a, its width CD at the base by h 

ertical height AF by c the inclination 

of it- outer face or extrados BC to the 

vertical by a„ that of its intrados AD by 

a,. 

Let II represent the centre of gravity of 
the parallelogram ADEB, and K that of 
the triangle BCE, and G that of the but- 
a : diaw 11 M, (xL, KX, perpendiculars 
upon AF. Then representing GL by X, 
and observing that the area ADEB is represented by ac, 
the area EBC by J(&— dfa and the area ADCB by ?(a+b)c, 

aL\lfiL + l(h-ayjns _ 2a.liM + (5-a)KX 
Ka+b)c ~ a + b 

Now ED£=HA+AM=±a+ii tan. a 7 =±(a + c tan. a,), 
KN==E2+fifc+JfcN=fi(5— a)+a+$ctaiL a 9 = 

±(b + 2a + '2c tan. a 3 ) ; 
Substituting these values and reducing, 
y _ ( a* + ab + 5 a ) + Q -f 25) c tan. a 2 
3(a + 6) 

&=CD=CF— DF=c tan. <*, + #— c tan. a 2 ; also (a* + ab + ?S) 

= {b— fl/ + ;y=6' 5 (tan.a- tan.a 2 ) 2 + 3tfc(tan. ai — tan.a,) + 3a 2 , 

(<z + 2£>)<?tan. a 2 = {2c (tan. «,— tan.a 2 )-f3#} c tan. a, 

=2<^ (tan. a,— tan. a,) tan. a^ + Sac tan. a, ; 
.\(a' + a£ + y) + (a + 2&)cftan. a 2 =:c 2 (tan.V— tan. 'a,) 
+ Zac tan. 0^ + 30*. 
. >*_ ^(tan. V,— tan. 3 a 2 ) + «<?tan. aj + a 8 
c(tan. a,— tan. a 2 ) + 2a 



. (413). 



• (414)- 



812. The Line of resistance in a buttress. 

Let LM represent any horizontal section of the buttress, 
TK a vertical line through the centre of 
gravity of that portion AMLB of the but- 
tress which rests upon this section. Pro- 
duce LM to meet the vertical AE in V, 
and let KV=X and AY=x ; then is the 
value of X determined by substituting x 
for o in equation (414). Let PO be the 
direction in which a single pressure P is 
applied to overturn the buttress. Take 

• ThJl equation is, of course, to be adapted to the case in which the inclina- 
khe other side of the vortical, as shown by the dotted line 
and therefere tan. a 2 negative. 




THE SLOPING BUTTRESS. 407 

OS to represent P in magnitude and direction, and ON tc 
represent the weight of the portion AMLB of the buttress ; 
complete the parallelogram BE, and produce its diagonal 
OR to Q ; then will OK evidently be the direction of the 
resultant pressure upon AMLB, and Q a point in the line of 
resistance. 

Let VQ=y, AG=Jc, /GrOT=t, f/.= weight of each cubic 
foot of material ; and let the same notation be adopted in 
other respects as in the last article. By similar triangles, 

QK_RI 

OK~ 01 

QK=QV-KV=y-\ 

OK=TK-TO=TK-TG cot. GOT=z-(x+&) cot. *, 

RI=R:N t sin. ENI==P sin. i, 

OI=OX+XI=iaAY(AB + LM)+EN cos. EjSI= 
%px{2a + x (tan. o^— tan. <*„)} + P cos. t ; 

y— X P sin, i 

" x— fc + k) cot. t"~^a?|2«+a?(tan. a,— tan. a a )}+P cos. C 

Transposing and reducing, 

_^i-x\2a-\-x (tan. a l — tan. a 2 )} + P (x sin. i— Jc cos. i m 
y~ ^33 j 2a + a? (tan. o^— tan. a 2 )} +P cos. t 

but substituting x for c in equation (414), and multiplying 
both sides of that equation by the denominator of the frac- 
tion in the second member, and by the factor ^x, we have 

i>^x{2a-\-x (tan. a t — tan. a a )} =}^x z (tan.v t — tan. \) + 
\^x ? a tan. a l + ^xd^ ; :.y= 

Svt/g 8 (tan.'a!— tan. a q 2 )4-^ a atan. «! -f-^/a;a 3 -j-2P(a; sin, c—k cos, i) 

fix 52a4-x(tan.a!— tan.a 3 )M-2P cos. i * ); 

which is the equation to the line of resistance in a buttress. 
If the intrados AD be vertical, tan. a 2 is to be assumed =0. 
If AD be inclined on the opposite side of the vertical to that 
shown in the figure, tan. a, is to be taken negatively. The 
line of resistance being of three dimensions in a?, it" follows 
that, for certain values of y, there are three possible values 
of x\ the curve has therefore a point of contrary flexure. 
The conditions of the equilibrium of the buttress are deter- 



WALL B1 STAINING THE 

mined from it- line of resistance precisely as those of the 

wall. 

Thus the thickness a of the buttress at its Bummit being 
given, and its height c, and it being observed that the dis- 
CE ifl represented by a + c tan. a„ the inclination a, of 
rtrados to the verticil may be determined, so that its 
line of resistance may intersect its foundation at a given dis- 
tance m from its extrados, by solving equation (415) in re- 
Bpect to tan. «,, having first substituted c for x and# + c tan. 
- _ m for y; and any other of the elements determining the 
conditions of the stability of the buttress may in like manner 
be determined by solving the equation (the same substitu- 
tions being made in it) in respect to that element. 



313. A WALL OF UNIFORM THICKNESS SUSTAINING THE PRES- 
SURE OF A FLUID. 

It' E be taken to represent the surface of the fluid, IK any 
section of the wall, and EP two thirds 
.__^_^_ the depth EK ; then will P be the cen- 
} tre of pressure* of EK, the tendency 
is§ of the fluid to overturn the portion 
%5f= AKIB of the wall being the same as 
would be produced by a single pressure 
applied perpendicular to its surface at 
P, and being equal in amount to the 
weight of a mass of water whose base 
55 " is equal to EK, and its height to the 
depth of the centre of gravity of EK, or 
to IV.K. Let AK= /. AE=«, weight of each cubic foot of 
the fluid=|Aj; 



- 






i ... P=(»-«).K«-«)f l =4{*-«)V. 

Let the direction of P intersect the axis of the wall in ; 
let it be represented in magnitude by OS; take OjS" to 
represent the weight of the portion AKIB of the wall ; com- 
the parallelogram SX, and produce its diagonal to 
meet EK in ( v ) ; then will Q be a point in the line of resist- 
L QM=y, AB=a, weight of each cubic foot of 

material of wall =f*. By similar triangles, !^- =^r. Now 

° 'MO iSO 

• Tr.Htisp on " Ilvdrostaties and Hvdrodvnaniics " bv the author of this 
work, Art. " 



PRESSURE OF A FLUID. 409 

QM=y, ~MO=YK=iEK=l(x-e), RN=OS=P== 

i^ x (x— ef, NO = weight of ABIK=^ax] 

" ±(x— e)~ pax ' ••* ■ ^ ax 

Dividing numerator and denominator of this equation by 

j*!, and observing that the fraction — represents the ratio a 

of the specific gravities of the material of the wall and the 
fluid, we have 

{co-ef_ 

9 6 oax K n 

which is the equation to the line of resistance in a wall of 
uniform thickness, sustaining the pressure of a fluid. 



314. To determine the thickness, a, of the wall, so that its 
height, h, being given, the line of resistance may intersect 
its foundation at a given distance, ra, within the extrados. 

Substituting, in equation (416), h for x, and \a—m for y, 
and solving the resulting equation in respect to a, we obtain 



a/ \ ( h ~ e Y 

a=m + V m + i^—T-- 



ah ■■■ (* 17 -) 

Equation (416) may be put under the form y= 

1 / e\ s 
77 — x* 1 1 — '-] ; whence it is apparent that y increases con- 
tinually with x ; so that the nearest approach is made by 
the line of resistance, to the extrados of the pier, at its 
lowest section, m therefore represents, in the above expres- 
sion, the modulus of stability (Art. 286). 



315. The conditions necessary that the wall should not be 
overthrown by the slipping of the courses of stones on on.e 
another. 

The angle SRO represents the inclination of the resultant 
pressure upon the section IK to the perpendicular ; the pro- 
posed condition is therefore satisfied, so long as SRO i6 less 
than the limiting angle of resistance p. 






THE 



= c7N 



08 B 



pax 



: the proposed con 






satisl i. so long m ., < tan. 9; or, 
- inequality, so long - 

»{i + (i+54 ,is.) 



•rO — fftf tan. 



^ ability of a wall of variable 
b staomro the pressure of a fluid. 



TRTCKHE89 



D J 


\ 

1 


1 


r=r^== 


/ 


MfflfE 


/ 

/ ■ 

- 






IHIHf 



Let us rirst suppose the internal face AB of the wall to be 
vertical : let XY be any a 
P the centre of pressure of EX. and 
SM a vertical through the centr 
gravity of the portion AXYD of the 
wall. Produce the horizontal direc- 
tion of the pressure P of the fluid, 

- 1 to be collected in its centre 

- ;re. to meet MS in S, and let 
e taken to represent it in m _ 

nitnd . STl - at the weight 

rti >n A X YT1 of the wall, 

te the parallelogram 8TRK; then will its 

sent the direction and amount of the 

- "■'■";- wore npon the nun AXYD. and if it be pro- 

mteiBGct XY in Q. Q will be a point in the line of 

.nee. 

k= . XQ=y, MX=>, AE=«, AD=a. inclina- 
rtical^a, ix = weight of cubical foot of wall, 
:' cubical : "fluid. Bv similar triangles, 

* 

SM~"ST" 

QM=QX-MX=y-\ SM=PX=iEX*=i(*— «); 
KT= fluid on EX=40.-jrx=W*— *)"f : 

ST= - ■ A.Y=H - :-::: tan. n\a*. 

The centre of pressure of a rectangular plane surface sustaining the 
pressure of a fluid is situated at two thirds the depth of its immersion.— 

." •' i- -.. ; . _••. 

of a heavy fluid on any plane surface is equal to the weight 
of a prism of the fluid whose base is equal in area to the surface pressed, and 

depth of the centre of gravitT of the surface pressed.— 
Hydro*tat%cs, Art. 31. y 



PEESSUKE OF A FLUID. 411 

y— \ i^i(^— e y 

Let— =o; then, if the fluid be water, o represents the 

specific gravity of the material of the wall ; and if not, it 
represents the ratio of the specific gravities of the fluid and 
wall. 



/ 1 V (x-ef 

~ y ~\Zaj2ax + x 2 ta 



/ 2ax + a? 2 tan. a * 

Now making a 2 =0 in equation (414), and substituting a 
for a l5 and a? for c, 

-Jar 2 tan. 2 a + ax tan. a + a 2 %x 5 tan. 2 a + ax* tan. a -f a?x 
a? tan. a + 2a 2ax-\-x 2 tan. a 

Adding this equation to the preceding, 

«-(#— £) 3 + -Ja? 3 tan.V-f&a? 2 tan. a + a a a? 

^~ 2aa?-j-a? 2 tan. a » • • \ J » 

which is the equation to the line of resistance to the wall, 
the conditions of whose stability may be determined from it 
as before (see Arts. 291. 293.). 



317. The conditions necessary ihat no course of stones com- 
posing the wall may slip upon the subjacent course. 

This condition is satisfied when the inclination of SQ to 
the perpendicular to the surface of contact at Q is less than 
the limiting angle of resistance 9 ; that is, when QSM <<p, 
or when 

tan. 9 >tan. QSM, or >^-, or >p _—,__^ . 

or tan. 9 > (- L j*~f .... (420). 
r ^ \oJ 2ax + x tan. a v ' 

No course of stones will be made by the pressure of the 
fluid to slip upon the subjacent course so long as this condi- 
tion is satisfied. 

It is easily shown that the expression forming the Becond 
member of the above inequality increases continually with 



Hi) Tin: STATURAL SLOPE OF EARTH. 

that the obliquity of the resultant pressure upon each 

ad the probability of its bring made to slip upon 

the next Bubjacenl course. Is greater in respect to the lower 

than the upper courses, increasing with the depth of each 

course beneath tin- surface of the fluid. 



Eabtii Works. 

318. The natural slope of earth. 

It has been explained (Art. 241.) that a mass, placed upon 
an inclined plane and acted upon by no other forces than its 

it and the resistance of the plane, will just be supported 
when the inclination of the plane to the horizon equals the 
Limiting angle of resistance between the surface of the plane 
and that of the mass which it supports; so that the limiting 
angle of resistance between the surfaces of the component 
parte of any mass of earth might be determined by varying 
continually the dope of its surface until a slope or inclination 

attained, at which particular slope 6mall masses of the 
Bame earth would only just be supported on its surface, or 
would just be upon the point of slipping down it. Now this 
process of experiment is very exactly imitated in the case of 
embankments, cuttings, and other earth-works, by natural 
causes, if a slope oi' earth be artificially constructed at an 
inclination greater than the particular inclination here 
spoken of, although, at first, the cohesion of the material 
may bo bind its parts together as to prevent them from slid- 

npon one another, and its surface from assuming its 
natural slope, vet by the operation of moisture, penetrating 

lass and afterwards drying, or under the influence of 
frost, congealing, and in the act of congelation expanding 
itself, this cohesion of the particles of the mass is continually 
in the process of being destroyed ; and thus the particles, so 
Long as the slope exceeds the limiting angle of resistance, 
are continually in the act of sliding down, "until, when that 
angle is at length reached, this descent ceases (except in so 
far as the particles continue to be washed down by the rain), 
and the surface retains permanently its natural slope. 

le limiting angle of resistance 9 is thus determined by 
observing what is the natural slope of each description of 
earth. 



THE PRESSURE OF EARTH. 



413 



The following table contains the results of some such 
observations * : — 



Natural Slopes of Different Kinds of Earth. 



Nature of Earth. 


Natural Slope. 


Authority. 


Fine dry sand (a single experiment) - 
Ditto 

Ditto - - ' - 
Common earth pulverised and dry - 
Common earth slightly damp - 
Earth the most dense and compact - 
Loose shingle perfectly dry 


21° 

34° 29' 

39° 

46° 50' 

54° 

55° 

39° 


Gadroy. 

Rondelet. 

Barlow. 

Rondelet. 

Rondelet. 

Barlow. 

Pasley. 



Specific Gravities of Different Kinds of Earth. 



[ 

Nature of Earth. 


Specific Gravity. 


Vegetable earth 

Sandy earth -------- 

Marl --------- 

Earthy sand -------- 

Rubble masonry of calcareous earth or siliceous stones 
Rubble masonry of granite - - - - - 

Rubble masonry of basaltic stones - 


1-4 
1-6 
1-9 
1-7 
1-7 to 2-3 
2-3 
2-5 



319. The pressure of earth. 

Let BD represent the surface of a wall sustaining the 
pressure of a mass of earth whose surface AE is horizontal. 

Let P represent the resultant of the pressures sustained 
by any portion AX of the wall; and let the cohesion of the 
particles of the earth to one another be neglected, as also 
their friction on the surface of the wall. It is evident that 



* It is taken from the treatise of M. Navier, entitled Resume (Tun Cours iU 
Construction, p. 160. 



TH1 ARTH. 



1 ■ [a 



Its deduced in respect to 
the dimensions of the wall, tl 



C — ■ 

— p • 



* 

the calculation being 
I, will be in . and 

the safe g 
Now the : earth which 

sees ponAX may yield in the 
stion of any oblique - 
XV. made from X to the surface 
i % AK of the mass. Suj • se VX * • 

>e the particular direction in which 
it actually tends to yield; s<> that 
-; ' : A X were removed, rupture 

=^ — ! LL * would first take place along this 

• d A X V be the portion of the mass which would 
tir>t fall. Then is the weight of the mass AYX supported 
by ti ft'erent element^ of the surface AX 
of the wall, v - Bultant is P. and by the resistance of 
the surface XI on which it tends to slide. Suppose, now, 
that the npon the point of sliding down the plane 
XV. the pressure P being that only which is just sufficient 

- ipport it: the resultant BK of the resistances of the 
different - XV is therefore inclined < Art. 241.) to the 

normal ST, at an angle RST equal to the limiting ang" 

- between any two contiguous suriaces of the 
earth. 

N ow the j.res>ure P. the weight TV <:>f the ma^s AXY. and 
the resistance P. being pressures in equilibrium, any two of 
them are t«» one another inversely as the sines of their incli- 
■ the third .Art. 14. i. 

. P _gk¥S B p sin. TVSP 

'"Wain. PSB ; " r - n Bbx psR- 

Put WSK=WST-RST=AYX— EST=~ «— 9, 

.\XY = <: P>P = P>T-PST=AXY-fPST=<-:. 

.'.{-^ - . . . . (421). 

AX . AY=^a,,c 5 tan. i ; if fx t =weight of each 
culA th, and AX=.r ; 

• V=uy- tan. 422). 

at that this expression, which represents 



THE PRESSURE OF EARTH. 415 

the resistance of the wall necessary to sustain the pressure of 
the wedge-shaped mass of earth AXY, being dependent for 
its amount upon the value of i (so that different sections, 
such as XY, being taken, each different mass cut off by such 
section will require a different resistance of the wall to sup- 
port it), may admit of a maximum value in respect to that 
variable.* And if the wall be made strong enough to supply 
a resistance sufficient to support that wedge-shaped mass of 
earth whose inclination t corresponds to the maximum value 
of P, and which thus requires the greatest resistance to sup- 
port it ; then will the earth evidently be prevented by it from 
slipping at any inclination whatever, for it will evidently not 
slip at that angle, the resistance necessary to support it at 
that angle being supplied ; and it will not slip at any other 
angle, because more than the resistance necessary to prevent 
it slipping at any other angle is supplied. 

If, then, the wall supplies a resistance equal to the maxi- 
mum value of P in respect to the variable *, it will not be 
overthrown by the pressure of the earth on AX. Moreover, 
if it supply any less resistance, it will be overthrown ; there 
not being a sufficient resistance supplied by it to prevent the 
earth from slipping at that inclination i which corresponds 
to the maximum value of P. 

To determine the actual pressure of the earth on AX, we 
have then only to determine the maximum value of P in re- 
spect to i. 

This maximum value is that which satisfies the conditions 

d? ft .<Z a P^ A 
-r-=0, and- JT <0. 
at at 

But differentiating equation (422) in respect to j, we obtain 
by reduction 

^ = fr y» in - 2 (,' + ?)-™-g ' .... (423)t 
dt * ' cos. h sin. 9 (i-h<p) v n 

Let the numerator and denominator of the fraction in the 

* The existence of this maximum will subsequently be shown : it is, how- 
ever, sufficiently evident, that, as the angle t is greater, the wedge-shaped r&Aafi 
to be supported is heavier; for which cause, if it operated alone, P would be- 
come greater as t increased. But as i increases, the plane XY becomes less 
inclined; for which cause, if it operated alone, P would become less as i in 
creased. These two causes thus operating to counteract one another, deter- 
mine a certain inclination in respect to which their neutralising influence is the 
least, and P therefore the greatest. 

\ Church's Diff. and Int. Cal., Art. 41. 



1 1»; no Ms. 

i<l member <»i* tin- equation be represented respectively 
bj y a id q ; therefore -rT=i ,J -^' . -A {'-<[ —' , -p\\ but when 

- =0, 0=0 ; in this case, therefore, -^rr=Tf x ,&'--r . Whence 

ih l at qdi 

it follows, by substitution, that for every value oft by which 
the firsl condition of a maximum is satisfied, the second dif- 
ferential co-efficient becomes 

P cos. 2(i + <p)-c o8. 2t 
'.It "* v cos. h sinT^) - I 4 *** 

Now it is evident from equation (423) that the condition 
-y-=0 is satisfied by that value of c which makes 2(* + ?)= 

*— 2t, or 



And if this value be substituted for t in equation (424), it 
;nes 

*? j r?^ 

C0S " (i-2) Sin - li+s) 

which expression is essentially negative, so that the second 
condition : - also satisfied by this value of t. It is that, there- 
fore, which corresponds to the maximum value of P; and 
substituting in equation (422), and reducing, we obtain for 
this maximum value of P the expression 

r=^'tan.'g-|)....(427); 

which expression represents the actual pressure of the earth 
on a Burface AX of the wall, whose width is one foot and its 
depth 



Revetement Walls. 
320. If, instead of a revetement wall sustaining the pres- 



REVETEMENT8. 



417 





A. X 


rVr 


| / 


1 i I 


^ | W / 


rrr 


-X l / 


i i 


^C\ ' / 


a* 1— T~ > 


r-">s 


tt 


j A % 


Vr 1 - 




yy- 





sure of a mass of earth, the weight 
of each cubic foot of which is re- 
presented by m* i5 it had sustained 
the pressure of & fluid, the weight 
of each cubic foot of which was re- 
presented by j* 1 tan. 2 (-— -J , then 

would the pressure of that fluid 
upon the surface AX have been 

represented* by J^sc'tan. 9 (----) 

so that the pressure of a mass of 
earth upon a revetement wall (equation 427), when its sur- 
face is horizontal (and when its horizontal surface extends, 
as shown in the iignre, to the very surface of the wall), is 
identical with that of an imaginary fluid whose specific gra- 
vity is such as to cause each cubic foot of it to have a weight 
f* 2 , represented in pounds by the formula 

,*,=,*, tan- '(|-|) (428); 

Substituting this value for ^ in equations (416) and (419), 
we determine therefore, at once, the lines of resistance in 
revetement walls of uniform and variable thickness, under 
the conditions supposed, to be respectively 

-tan. 2 1 ~_| J(x— ef +-£o?'tan.'a + asc'tan.a -f a\ 



y= 



So 



2ax + cc 2 tan. a 



..(430); 



where a represents the ratio of the specific gravity of the 
material of the wall to that of the earth. The conditions of 
the equilibrium of the revetement wall may be determined 
from the equation to its line of resistance, as explained in 
the case of the ordinary wall. 



27 



* Hydrostatics, Art. 31. 



41$ REYETOff* 

' n I,., the limiting angle of distance 
to distinguish it from 



tan. o.>^ ,an - 



■6-3 



— 



— *' tan. a 



431 : 



where . rer-sent, the ratio of the specific gravity of the 
in the lower courses than the lugher. 

,e surface of re CW m ^femh.TX 

rnpr the ma- 

contact with the surface 
AX : a revetement wall 
a | -ake place AX= 
x AXY = '. xa: = 

if ^V be taken to 

represent the weight of 

the maffl AXT. it inav be 

a in Art. 319, 

equation Out V- 

1 W c L (*+?> 




: - AX . AY 



ay ,•:.. -:. AY= 



___; *~ 



sin 

tfm _ * _ • 

fore Wr - _ . - T~ 

-* the value of. in this function is that which renderc 
lialimu; -ding cot. .i-=> and dif 



REVETEMENT6. 



419 



ferentiating in respect to tan. i, this value of i is readily 
determined to be that which satisfies the equation 

cot. i=tan. 9 + sec. 9 Vl + cot. /3 cot. 9 (433). 

Substituting in equation (432), and reducing, 

P =^ 2 i ^ V U w 

(1-f sin. 9 4/1 + cot. 9 cot. £) V ; * 

From which equation it is apparent, that the pressure of the 
earth is m this case, identical with that of a fluid, of such a 
density that the weight ^ of each cubic foot of it, is repre- 
sented by the formula F 



'Hi 



cos. 9 



+sin. 9 4/l+cot.pcot./3 



r 



(435). 



The conditions of the equilibrium of a revetement wall 
sustaining the pressure of such a mass of earth are therefore 

(Arts^fanl %%"" ^^ " *~ "** ** ^ 



323. The Resistance of Earth. 

Let the wall BDEF be supported by the resistance of a 

mass of earth upon its sur- 



face AD, a pressure P, ap- 
plied to its opposite face, 
tending to overthrow it. Let 
the surface AH of the earth 
be horizontal ; and let Q 
represent the pressure which, 
being applied to AX, would 
just be sufficient to cause the 
mass of earth in contact 
with that portion of the wall 
to yield; the prism AXY 
llrf _ YV slipping backwards upon the 
surrace aY. Adopting the same notation as in Art 310 
ana proceeding in the same manner, but observing that RS 
is to be measured here on the opposite side of TS (Art 24:1) 
since the mass of earth is supposed to be upon the point of 
slipping upwards instead of downwards, we shall obtain 
Qz^^aj* tan. 1 cot. (1— 9) (436). 




420 



WALLS BACKED BY EAETH. 



Now it is evident that XY is that plane along which rup- 
ture may be made to take place by the least value of Q ; i 
in the above expression is therefore that angle which gives 
to that expression its minimum value. Hence, observing 
that equation (436) differs from equation (422) only in the 
sign of <p, and that the second differential (equation 426) is 
rendered essentially positive by changing the sign of 9, it is 
apparent (equation 427) that the value of Q necessary to 
overcome the pressure of the earth upon AX is represented 

by 

Q=** l rftan. , g+!) ....(437). 



324. It is evident that a fluid would oppose the same 
resistance to the overthrow of the wall as the resistance of 
the earth does, provided that the weight m- 4 of each cubic 
foot of the fluid were such that 



fV 



:Man.*(j+!) ....(438); 



so that the point in AX at which the pressure Q may be 
conceived to be applied, is situated at f ds the distance AX. 



325. The stability of a wall of uniform thickness which a 
given "pressure P tends to overthrow, and which is sus- 
tained by the resistance of earth. 

Let y be the point in which any section XZ of the wall 

would be intersected by the 
resultant of the pressures 
upon the wall above that sec- 
- r __._T.._ tion, if the whole resistance 
" * /^'f Q, which the earth in con- 
,/' ;| tact with AX is capable of 
if supplying, were called into 
>i action. Let BX=a?, Xy=y, 
I BA = e, BE=«, Bp = k, 
g weight of cubic feet of ma- 
j terial of wall=,a, inclination 
f: of P to vertical =6. Taking 
&5s*Mtf--. || ie momen ts about the point 
y of the pressures applied to BXZE, we have, by the prin- 
ciple of the equality of moments, observing that XQ=| 




•* /—&£ 



WALL6 BACKED BY EARTH. 



421 



(x— e), and that the perpendicular from y, upon P is repre- 
sented by x sin. &—(k—y) cos. 0, 

Y\x sin. &—Qc—y) cos. &\ =%(x—e)Q + (ia—yyax; 

or substituting for Q its value (equation 437), and solving in 
respect to y, 

-^ i (x—ey-\-^a 2 x—Y(x sin. b— Tc cos. 6) 



y- 



(439). 



P cos. Q+^ax 

Now it is evident that the wall will not be overthrown 
upon any section XZ, so long as the greatest resistance Q, 
which the superincumbent earth is capable of supplying, is 
sufficient to cause the resultant pressure upon EX to inter- 
sect that section, or so long as y in the above equation has 
a positive value ; moreover, that the stability of the wall is 
determined by the minimum value of y in respect to x in 
that equation, and the greatest height to which the wall can 
be built, so as to stand, by that vahce of x which makes y=0. 



326. The stability of a wall which a given pressure tends to 
overthrow, and which is supported by a mass of earth 
whose surface is not horizontal. 



Let 




represent the inclination of the surface AB of earth 
to the horizon. By reasoning 
similar to that of Art. 322., it is 
apparent that the resistance Q 
of the earth in contact with any 

fiven portion AX of the wall to 
isplacement, is determined by 
assigning to <p a negative value 
in equation (434). Whence it 
follows, that tliis resistance is 
equivalent to that which would 
be produced by the pressure of 
'V\ a fluid upon the wall, the weight 
^^IxSfSS^c'flti^ |u- 5 of each cubic foot of whicli 
was represented by the formula 



','•;- \ 



''&S 



"Hi- 



cos. 9 



sin. <p yl— cot. <p cot 



*!■ 



(440). 



The conditions of the stability of an upright wall sub- 
jected to any given pressure 1* tending to overthrow it, and 



422 



EEVETEMENT8. 



ined bj the pmmwiffl of such a mass of earth, are there- 

prectaeij the same as those discussed in the last article; 

the Bymbol f* 4 (equation 439) being replaced by f* § (equation 



\.V a 



837. Tht stability of a revetement wall whose interior face 
netined to im vertical at any angle / taking into account 
th< friction of the earth ujpon the face of the wall. 

Lit «.. represent the inclination of the face BD of such a 
Avail to the vertical, <p, the limiting angle of resistance 
between the mass of earth and the surface of the wall; and 
let the same notation be adopted as in the last article in 
respect to the other elements of the 
question, and the same construction 
made. Draw PQ perpendicular to BD ; 
then is the direction PS of the resist- 
ance of the wall upon the mass of earth, 
evidently inclined to QP at an angle 
QPS equal to the limiting angle of 
it resistance <p„ in the state bordering 
J upon motion by the overthrow of the 
% wall* (Art. 241.). 

^ Draw Yn horizontally and Xa verti- 
£2&£l+j} oallj, produce TS and KS to meet it in 
m and w, and let aXY= i, 

P _ sin. WSB sin. (WST-TSR ) 
'' W ~ sin. PSP ~ sin. (KmP + SPw). 



±3? bs '/ 







Bat WOT=ATX=- 



aXY=-- h TSH=<p, 

Ri»P=TnP+wS»=<?XT+EST=t+^ 
SPw»SPQ+QP«=sip w +« t ; 

.p. gini lr t - > ) _ cos.(, + ,) 

W sin. (« + <p + <p, + a 9 ) sin.(t + a, + <p + <p 2 )> 
Also Ws^aX . AY=if* 1 aj'(tan.£ + tan. a„) ; if aX=x, 



It is not only in the state of the wall bordering upon motion that thU 
direction ot' the resistance obtains, but in every state in which the stability of 
tht wall is maintain. -d. (S.c (he PrincipU of Least Resistance.) 



REVETEMENT8. 4:23 

^ 8 in.(< + a 2 + 9+<P 2 ) v ; * 

Assuming a,-f<p + ?,=£, then differentiating in respect to *, 

dP 

and assuming -j- = 0, we obtain b y reduction 

—(tan. t-f tan. a 2 ) cos. (/3— 9) + 

cos. (4+9) sin. (t-f/3) sec. a t=0 ; or, 

— (tan. 1 + tan. a 2 ) (1 4- tan. /3 tan. 9) -f- 

(1— tan. 1 tan. 9) (tan. t + tan. /3)=0 ; 

,\ tan. 9 1 -f- 2 tan. * tan. (3 — tan. £ cot. 9 + 

(cot. 9 + tan. /3) tan. a 3 = 0. 

Solving this quadratic in respect to tan. t, neglecting the 
negative root, since tan. 1 is essentially positive, and reducing, 

tan. f=(tan. (3— tan. a 2 )*(tan. /tf-f-cot. 9)*— -tan. /3 . . . (442.) 

ISTow the value of 1 determined by this equation, when 
substituted in the second differential coefficient of P in 
respect to 1, gives to that coefficient a negative value ; it 
therefore corresponds to a maximum value of P, which 
maximum determines (Art. 319.) the thrust of the earth 
upon the portion AX of the wall. To obtain this maximum 
value of P by substitution in equation (441), let it be 
observed that 

cos. (i-f <p)_l— tan. 1 tan. 9 /cos. 9 \ 

sin. (t+P)~ (tan. i + tan. /3) \cos. /3'/ 

1— tan. 1 tan. 9=1 + tan. (3 tan. 9— tan. 9 (tan. (3 — 

tan. a 2 )*(tan. /3-fcot. 9)*, 

=tan. 9 (tan. /3 + cot. 9)* {(tan. £-f cot. 9)*— (tan. (3— tan. a 2 )*} 

tan. * + tan. /3=(tan. /3 + cot. <p)*(tan. (3— tan. a 2 )*; 

< cos. (/+<?)_ sin. 9 j /tan. /3 + cot. 91* ) 

"sin. (* + /3) = cos. # ( \tan. /3— tan. a J ~~ J * 

Also tan. t + tan. a 2 =(tan. (3-\-cot. 9)*(tan. /3— 

tan. a 2 V— (tan. (3— tan. a 2 ) 

—(tan. /3— tan. a 2 )*j(tan. /3-fcot. 9)*— (tan. (3— tan. a,)*}, 

sin 9 
.•.P=| f x 1 a? a --^K tan - + cot. 9) f -(tan. 0— tan. a,)*}*; 









mder the following form ; 
_ .rithmic calculation, 

a- : -- - : -^. >' ' ' 

1 r - -"".. - ' 1 cob.., / I ■ 

ts value a. — - — - . 



F 



•■■ --- 



. 443. 



Bj a com} " this equation with equation 

:it, that the ae . arth upon a 

under the £ ifi identical 

with that whi if it w^re perfectly fluid, 

;h cubi- hat fluid had 

: _ ' in the above 
that the condit: - - : .eh a 

:.t wall are identical (this value being supp 
with • stability of a wall sustaining the 

.eept that the pressure of the earth is 
the wall in a direction perpendicular * 
: a fluid is. but in a direction inclined to 
the perpendicular at a given angle, namely, the limiting 
ang. oe. 



_ - The raa p earth which sup.m nwa a kevete- 

ME>T WALL A>~D SLOPES I TO -UMMLT. 







rto we have supposed the surface of the earth whose 
pr - a sustained by a revete- 

ment wall to be horizontal ; let us 

- surface to be ele- 
the summit of the wall, 

and : .rural 

all is then said to be 

r to carry a parapet. 

-ent the natural slope 

of the ear: h. FT its horizontal sr.r- 

tace, BX any :' the internal 

- f the wall, P the 
•ntal pressure just necessarv 






REVETEMENTS. 425 

to support the mass of earth HXYF, whose weight is "W", 
upon the inclined plane XY. Produce XB and YF to meet 
in A, and let AX=x, ATL=c, AXY=l, ^ l = weight of each 
cubic foot of the earth, 9 the natural slope of its surface 
FE. Now it may be shown, precisely by the same reason- 
ing as before, that the actual pressure of the earth upon the 
portion BX of the wall is represented by that value of P 
which is a maximum in respect to the variable t ; moreover, 
that the relation of P and 1 is expressed by the function P 
—W cot. (* + <p); where W=^(area HXYF)=^(AXY - 
AHF)=^ 1 (|^ 2 tan. 1— ic 2 cot. 9) ; 

.\P=ifA x (fl? tan. i—c 2 cot. <p) cot. (4+9) (444). 

Expanding cot. (*+<p), 

_ x (a? 2 tan. 1— & cot. 9) (1— tan. 1 tan. 9) 
2,1X1 ~ tan. l + tan. 9 

To facilitate the differentiation of this function, let 
tan. t + tan. 9 be represented by s, and let it be observed 
that whatever conditions determine the maximum value of P 
in respect to z determine also its maximum value in respect 
to 1* Then tan. i=z— tan. 9 ; therefore 1 — tan. 1 tan. 9= 
1—2 tan. 9 + tan. 2 9= — z tan. 9 + sec. 2 9. Also, a? 2 tan. <— 
c* cot. cp=x*z— (x* tan. 9 + c 2 cot. 9). 

Substituting these values in the preceding expression for 
P, and reducing, 

( ( x* tan. 9 + c 2 cot. 9 ) sec. 3 9 , 
F=ift; i —zx tan. 9— -f 

a? 2 (sec. 2 9 + tan. 3 9) + c 2 1 (445). 

d¥ f (aj 3 tan. 9 + c 2 cot. 9) sec. 9 9 ) 

•"•"& =*^ 1 -^ tan -?+ 31 — 1 » 



sec. % there- 



^ ^ tfP <7P<k , <FP <PP/dz \» . dP tf a 2 dz 

For *=& A' and W = ^[a ) + Tz d7' now -dc 

fore -=-=-=- sec. 2 i: and for all values of t less than -, sec. a t has a finite 
dc dz 2' 

value, so that -.- = when -^-=0. 
di dz 

When, moreover, -r-=0. - V7r = — rs- It-); so that, when -,^-is negative, 
1 dz dt 2 dz 2 \di ) dz 2 

~r~ Is also negative. 



4'26 K! VKIKMKNTS. 

tTP (x* tan. <p + (? cot. 9) sec. '9 

Tht fiisl condition of a maximum is therefore satisfied by 

. quation 

(x'tan.^ + c'cot.^sec.^ 
-^tan.v-K ^ =0 (446;; 

►lving this equation in respect to 3, and reducing, it is 
pfied by the equation 

2= ± I sec. 3 9 4- — cosec. a 9J . 

\ >w the second condition of a maximum is evidently 
satisfied by any positive value of 2, and therefore by the 
positive root 01 this equation. Taking, therefore, the posi- 
tive sign, substituting for z its value, and transposing, 

tan. 1= I sec. 3 9 + — a cosec. "9 J —tan. 9 (447) ; 

w hich equation determines the tangent of the inclination 
AX Y to the vertical, of the base XY of that wedge-like 
mass of earth IIXYF, whose pressure is borne by the sur- 
face I»X <>i* the wall. To determine the actual pressure 
upon the wall, this value of tan. 1 must be substituted in the 
expression for P (equation 445). Kow the two first terms 
of the expression within the brackets in the second member 
of that equation may be placed under the form 

j , (a? 3 tan. 9 -he 3 cot. 9) sec. "9 ) 
—z j a? tan. <p + - ^ I . 

But it appears by equation (446) that the two terms which 
compose this expression are equal, so that the expression : 
equivalent to — 2zx* tan. 9 ; or, substituting for the va 1 

g, to — -_>./•' tan. 9 (sec. 3 9H — j cosec. V)*, or to —2x se 

X 

;n. > -f c 5 )*. Substituting in equation (445), 

P=-i.^ I j-*- > ./'see. c >Vtan.^ + c a )*+(a?tan. a 9 + c 3 ) + a? sec '9} 
.-. Pstfajfl sec. 9-(a 3 tan. 3 9 + cV ^ (448); 

by which expression is determined the actual pressure upon 
a portion <>\' the wall, the distance of whose lowest point 
from A is represented by x. 



REYETKMEXTS. 



427 



4~* 



329. The conditions necessary that a revetement wall carry- 
ing a parapet may not be overthrown by the slipping of 
any course of stones on the subjacent course. 

Let pj represent the limiting angle of the resistance of the 
stones of the wall upon one another ; and let OQ represent 
the direction of the resultant pressure 
on the course XZ. The proposed 
\ conditions are then involved (Art. 
.} 141.) in the inequality p^QOM, or 
J j tan. <p a > tan. QOM, or tan. <p > 

| OS ^veightofBZ* "' or substituting 
:? for P its value (equation 448), and 
£ %^(2ax+x* tan. a) for the weight of 
I BZ, it appears that the proposed 
| conditions are determined by the 
- inequality 




tan. <p,> 



/f*! \x sec. 



<p— (aftan. , <p + c")*} 1 ] 



2ax + x tan. a 



(449). 



330. The line of resistance in a revetement wall carrying a 

parapet. 

Let OT be taken to represent the pressure P, and OS the 
weight of BZ. Complete the parallelogram ST, and pro- 
duce its diagonal OR to Q ; then will Q be a point in the 
line of resistance. Let AX=x, QX=y, AB=£, AP=X, 

XM=X, ~W = weight of BZf. By similar triangles, 7TTr= 
5| ; but QM=(t/-X), OM=a?-X, RS=P, OS=W; 



y— X P 

x-x~vr 



v- 



Wx+IV-PX 



(450). 



Now the value of X is determined from equation (414), by 



* The influence, upon the equilibrium of the wall, of the small portion of 
earth BIIE is neglected in this and the subsequent computation. 

f The influence of the weight of the small mass of earth BEII which rests 
on the summit of the wall is here again neglected. 



\ l.II M i:\T3. 

substituting in that equation {x — b) for c: whence we obtain, 
.- tan. a,=0, and substituting a for a„ 

__ £(#— £>) a tan. a a + fl(a;— ft) tan, a + a* . 
(x—b) tan. a + 2a 

Ako W=JK»-&)K*-*) tan -*+ 2a } ( 451 )5 

• WX=Jf*(aj— J)ji(flj— J) 1 tan. \ + a{x-l) tan. a -f a 1 } . 

It remains, therefore, only to determine the value of the 
term P . X. Now it is evident (Art. 16.) that the product 
P . X is equal to the sum of the moments of the pressures 
upon the elementary surfaces which compose the whole sur- 
face I'X. But the pressure upon any such elementary sur- 
istance from A is 2, is evidently represented 

by '-, -X/-" :: '; its moment is therefore represented by —^xAx, 

and the sum of the moments of all such elementary pressures 

by ~ , 'A.r, or when a# is infinitely small, by 

/dx: therefore P . X= / -^—xdx. 
dx J dx 

b b 

But differentiating equation (448), 

dF . . , 9 , ., ( x tan. 2 o ) 

-T-=f*i K sec. o—(,v tan. ? + c a W < sec. 9— ttt — i — r~>u > • 
dx ' ( v y > / (a? tan. 9 + <ry \ 

Performing the actual multiplication of the factors in the 

ad member of this equation, observing that ^ =- — 5-7 

H & (a: a tan. s 9 + c 9 )* 

(^t an.^H-c')-c 3 ,..,,. o a , 

= / , . § — , 3li , = U' tan. y + c |* — 7-=- 5 — ■ — =r\, and re- 

tan.- y + r-,» Y J (^tan.^ + c 3 )^ 

ducing we obtain 

* V being a function of r, lot it be represented by f(x)\ then will f{x) repre- 
sent ti upon ■ portion of the surface BX terminated at the distance 
x from A, and flz-^-Ax) that upon a portion terminated at the distance x-\-Ax\ 
■-■-f-Ajr) — fz will represent the pressure upon the small element Ax 
of the surface included between those two distances. But by Taylor's theorem, 

/(*+A«)— /*= A* -f- -j a \t^~4-> & c - 5 therefore, neglecting terms in 
solving powers of Ax above the first, pressure on element = -=- Ax. 



THE ARCH 



429 



^===^|a:(sec. 2 9 + taii. a 9)--2sec.9(^taii. 2 9 + c 2 > + 



c sec. 9 



(x'tsin.^ + cy 



Multiplying this equation by a?, and integrating between the 
limits b and x, 



'•J(sec. 2 cp+tan. 2 <p)(cc 3 — J 8 )— fsec. <p cot. *<p{(a? tan.V 
P.X=mJ +c 2 )f —Qf tan. 2 <p + c 2 )f } +c 2 sec. 9 cot. 2 9 

{{x 2 tan. 2 9 + ^-(J 2 tan. 2 (p + c 2 )^ (452). 

This value of P . X being substituted in equation (450), 
and the values of TVX 5 W, P, from equations (448) and 
(451), the line of resistance to the revetement wall will be 
determined, and thence all the conditions of its stability 
may be found as before.* 



The Arch. 

331. Each of the structures, the conditions of whose sta- 
bility (considered as a system of bodies in contact), have 
hitherto been discussed, whatever may have been the pres- 
sures supposed to be insistent upon it, has been supposed to 
rest ultimately upon a single resisting surface, the resultant 
of the resistances on the different elements of which was at 
once determined in magnitude and direction by the resultant 
of the given insistent pressures! being equal and opposite 
to that resultant. 

The arch is a system of bodies in contact which reposes 
ultimately upon two resisting surfaces called its abutments. 
The resistances of these surfaces are in equilibrium with the 

* The limits which the author has in this work imposed upon himself do not 
leave him space to enter further upon the discussion of this case of the 
revetement wall, the application of which to the theory of fortification is so 
direct and obvious. The reader desirous of further information is referred to 
the treatise of M. Poncelet, entitled " Memoire sur la Stability des Revete- 
ments et de leura Fondations." lie will there find the subject developed in all 
its practical relations, and treated with the accustomed originality and power 
of that illustrious author. The above method of investigation has nothing in 
common with the method adopted by M. Poncelet except Coulomb's principle 
of the wedge of maximum pressure. 

f The weight of the structure itself is supposed to be included among these 
pressures. 



4. it THK PKrNCIPLE OF LEAST RESISTANCE. 

stent upon tlie arch (inclusive of its 

unt of the resultant 

■ lent npoo the unknown 

site Banace :hns the general 

e determin; the line of 

iditioofl : stability, in that 

which repose on a single resisting 

- in the case of the arch. 



The presxiple of least re- 

art a g -sea 

briu 

system, which are not resista: 

be represented by A. and the resistances by B : also let any 

others - suret vhich may be made to replace the 

s B ind sustain A. be represented by ( 

Suppose the system B to be replaced by C : then it is 

rent that each pressure of the system C is equal to the 

_ :ed to its point of application from the 

.ual to that pressure, 

with the pr h bo propagated to it from the 

b the system C. In the former case it is 

sisl 3 of 1 - stem B : in the 

9e it is greater than it. Hence, therefore, it appears 

- re oi the system B is a minimum, subject 

ic conditions imy the eqirlibriuin of the whole. 

of the pressures applied to a body, other 

than the resistant s, 1 ident from the a 

- are the so as t - istain 

ind therefore that if each resisting point be 

stance in any d ... then are 

all t. .ne another and to the result- 

f the other prest i applied to the b 

• The principle of least resistance was first published bv the author of thi« 
work in the Philosophical Magazine for October, 1 



THE ARCH. 431 

333. Of all the pressures which can he applied to the highest 
voussoir of a semi-arch, different in their amou its and 
jpoints of application, hut all consistent with the equili- 
brium of the semi-arch, that which it would sustain from 
the pressure of an opposite and equal semi-arch is the least. 

Let EB represent the surface by which an arch rests upon 
¥ 




either of its abutments ; then are the resistances upon the 
different points of that surface (Art. 331.) the least pressures, 
which, being applied to those points, are consistent with the 
equilibrium of the arch. They are, moreover, parallel to one 
another : their resultant is therefore the least single pressure, 
which, being applied to the surface EB, would be sufficient 
to maintain the equilibrium of the arch, if the abutment were 
removed. 

Now, if this resultant be resolved vertically and horizon- 
tally, its component in a vertical direction will evidently be 
equal to the weight of the semi-arch : it is therefore given in 
amount. In order that the resultant may be a minimum, its 
vertical component being thus given, it is therefore necessary 
that its horizontal component should be a minimum ; but 
this horizontal component of the resistance upon the abut- 
ment is evidently equal to the pressure P of the opposite 
semi- arch upon its key-stone : that pressure is therefore a 
minimum ; or, if the semi-arches be equal in every respect, 
it is the least pressure which, being applied to the side of the 
key-stone, would be sufficient to support either semi-arch ; 
which was to be proved. 

The following proof of this property may be more intelli- 
gible to some readers than the preceding. It is independent 
of the more general demonstration of the principle of least 
resistance.* 

* See Memoir by the author of this work in Mr. Hann's " Treatise on the 
Theory of Bridges,'" p. 10. 



132 



1111. ARCH. 



'Hi.- pressure which an opposite semi-arch would produce 
upon the Bide AD of the key-stone, is equal to the tendency 
ut' that Bemi-arch t<> revolve forwards upon the interior edges 
of one or more of it- voussoirs. Now this tendency to motion 
idently equal to the least force which would >upportthe 
Bemi-arch. If the arches be equal and equally 
loaded, it i.- therefore equal to the least force which would 
Bupporl the Bemi-arch ABED. 



334. General conditions of the stability of an arch.* 




Suppose the mass ABDC to be acted upon by any number 

of pressures, among which 
is the pressure Q, being the 
resultant of certain resist- 
ances, supplied by different 
points in a surface BD ; 
common to the mass and to 
an immoveable obstacle 
BE. 

Now it is clear that un- 
der these circumstances we 
may vary the pressure P, 
both as to its amount, di- 
rection, and point of appli- 
cation in AC, without disturbing the equilibrium, provided 
only the form and direction of the line of resistance continue 
t«> Batisfy the conditions imposed by the equilibrium of the 
em. 
These have been shown (Art. 283) to be the following : — 
that it no where cut the surface of the mass, except at P, 
and within the space BD ; and that the resultant pressure 
upon no Bection MX of the mass, or the common surface BD 
of the mass and obstacle, be inclined to the perpendicular to 
that Burface, at an angle greater than the limiting angle of 
resistance. 

r llms, varying the pressure P, we may destroy the equi- 
librium, either, first, by causing the resultant pressure to 
take a direction without the limits prescribed by the resist- 
ance of any Bection MX through which it passes, that is, 
without the cone of resistance at the point where it inter- 

* Theoretical ami Practical Treatise on Bridges, vol. i. ; Memoir by the au- 
thor of this work, p. 11. 



THE AKCB. 433 

sects that surface ; or, secondly, by causing the point Q to 
fall without the surface BD, in which case no resistance can 
be opposed to the resultant force acting in that point ; or, 
thirdly, the point Q lying within the surface BD, we may 
destroy the equilibrium by causing the line of resistance to 
cut the surface of the mass somewhere between that point 
and P. 

Let us suppose the limits of the variation of P, within 
which the first two conditions are satisfied, to be known ; and 
varying it, within those limits, let us consider what may be 
its least and greatest values so as to satisfy the third condition. 

Let P act at a given point in AC, and in a given direc- 
tion. It is evident that by diminishing it under these 
circumstances the line of resistance will be made continually 
to assume more nearly that direction which it would have 
if P were entirely removed. 

Provided, then, that if P were thus removed, the line of 
resistance would cut the surface, — that is, provided the 
force P be necessary to the equilibrium, — it follows that by 
diminishing it we may vary the direction and curvature of 
the line of resistance, until we at length make it touch some 
point or other in the surface of the mass. 

And this is the limit ; for if the diminution be carried 
further, it will cut the surface, and the equilibrium will be 
destroyed. It appears, then, that under the circumstances 
supposed, when P, acting at a given point and in a given 
direction, is the least possible, the line of resistance touches 
the interior surface or intrados of the mass. 

In the same manner it may be shown that when it is the 
greatest possible, the line of resistance touches the exterior 
surface or extrados of the mass. 

The direction and point of application of P in AC have 
here been supposed to be given ; but by varying this direc- 
tion and point of application, the contact of the line of 
resistance with the intrados of the arch may be made to 
take place in an infinite variety of different points, and each 
such variation supplies a new value of P. Among these, 
therefore, it remains to seek the absolute maximum and 
minimum values of that pressure. 

In respect to the direction of the pressure P, or its incli- 
nation to AC, it is at once apparent that the least value of 
that pressure is obtained, whatever be its point of applica- 
tion, when it is horizontal. 

There remain, then, two conditions to which P is to be 
subjected, and which involve its condition of a minimum. 

28 



434 



THE ARCH. 



The first is, t : to the 

fith the intrados / the 
■d, thai ' keystone AO 

thai' 

On. 



335. Practical conditions of the stability of an- arch 
of uncemented stunls. 

The condition, however, that the resultant pressure upon 
the k is subject, in respect to the position of its 

application on the key-stone, to the condition 
minimum, is dependent upon hypothetical qualities of the 
masonry. It supposes an unyielding material for the arch- 
a, .da mathematical adjustment of their surface-. 
ristence in the uncemented arch. On the 
striking ot' the centres the arch invariably sinks at the 
crown, its - - there slightly opening at their I 

_ -. and pressing upon one another exclusively by their 
upper edges. Practically, the line of resistance then, in an 
arch • " . touches the extrados at the crown ; 

t «> that only the first of the two conditions of the minimum 

actually obtains : that, namely, which giv 
the li tance a contact with the intrados of the 

arch. This condition being assumed, all consideration of 
the yielding quality of the material of the arch and its 
abutments - inated. 

Th- :' the solid has hitherto been assumed to "be 

. together with the positions of the different sections 

made through it : and the forms of its lines of resistance and 

- ire. and their directions through its mass have thence 

irmined. 
It is 3t that the converse of this operation is pos- 

H 8 _ _ en the form and position of the line of resist- 
ance or of pressure, and the positions of the different sections 
: iade through the mass, it may. for instance, be 
inquired what form these conditions impose upon the surface 
which bounds it. 

the dire -he line of resistance or pressure and 

_ -:irfaee may be subjected to certain 

determining either. 

:''»r instance, the form of the intrados of an arch be 

1 the d:: :' the intersecting plane be always 






THE AKCH. 435 

perpendicular to it, and if the line of pressure be supposed 
to intersect this plane always at the same given angle with 
the perpendicular to it, so that the tendency of the pressure 
to thrust each from its place may be the same, we may 
determine what, under these circumstances, must be the 
extrados of the arch. 

If this angle equal constantly the limiting angle of resist- 
ance, the arch is in a state bordering upon motion, each 
voussoir being upon the point of slipping downwards, or up- 
wards, according as the constant angle is measured above or 
below the perpendicular to the surface of the voussoir. 

The systems of voussoirs which satisfy these two con- 
ditions are the greatest and least possible. 

If the constant angle be zero, the line of pressure being 
every where perpendicular to the joints of the voussoirs, the 
arch would stand even if there were no friction of their sur- 
faces. It is then technically said to be equilibriated ; and 
the equilibrium of the arch, according to this single con- 
dition, constituted the theory of the arch so long in vogue, 
and so well known from the works of Emerson, Sutton, and 
Whewell. It is impossible to conceive any arrangement of 
the parts of an arch by which its stability can be more 
effectually secured, so far as the tendency of its voussoirs to 
slide upon one another is concerned: there is, however, 
probably, no practical case in which this tendency really 
affects the equilibrium. So great is the limiting angle of 
resistance in respect to all the kinds of stone used in the 
construction of arches, that it would perhaps be difficult to 
construct an arch, the resultant pressure upon any of the 
joints of which above the springing should lie without this 
angle, or which should yield by the slipping of any of its 
voussoirs. 

Traced to the abutment of the arch, the line of resistance 
ascertains the point where the direction of the resultant 
pressure intersects it, and the line of pressure determines the 
inclination to the vertical of that resultant ;* these elements 
determine all the conditions of the equilibrium of the abut- 
ments, and therefore of the whole structure; they associate 
themselves directly with the conditions of the loading of the 
arch, and enable us so to distribute it as to throw the points 
of rupture into any given position on the intrados, and give 
to the line of resistance any direction which shall best con- 

* The inclination of the resultant pressure at the springing to the vertical 
may be determined independently of the line of pressure, as will hereafter be 
shown 



43G THE LINE OF RESISTANCE IN THE ARCH. 

duce to the stability of the structure; from known dimen- 
sions, and a known Loading of the arch, they determine tho 
dimensions of piers which will support it; or conversely, 
from known dimensions of the piers they ascertain the 
dimensions and loading of the arch, which may safely be 
made to Bpan the space between them. 



336. To DETERMINE THE LTNE OF RESISTANCE TN AN ARCH 
WHOSE TNTRADOS 18 A CIRCLE, AND WHOSE LOAD IS COL- 
I.l< ll.li OYER TWO POINTS OF ITS EXTRADOS SYMMETRICALLY 
PLACED IN RESPECT TO THE CROWN OF THE ARCH. 

Let ADBF represent any portion of such an arch, F a 

pressure applied at its extreme 
j 5 " D voussoir, and X and Y the ho- 

I j^j -«**—;* rizontal and vertical compo- 

3ca »---~^H ( ^A nents of any pressure borne 

'V^jy j upon the portion DT of its ex- 

/5v?x trados, or of the resultant of 

~f\y S ^ any number of such pressures ; 

I \ let, moreover, the co-ordinates, 

■ 1 .. \ from the centre C, of the point 

' '— ' \ of application of this pressure, 

\ or of this resultant pressure, be 

\j x and y. 

Let the horizontal force P 
be applied in AD at a vertical distance p from C ; also let 
( T represent any plane which, passing through C, intersects 
the arch in a direction parallel to the joints of its voussoirs. 
Let this plane be intersected b} T the resultant of the pres- 
Bnres applied to the mass ASTD in R. These pressures are 
the weight of the mass ASTD, the load X and Y, and the 
jure I'. Now if pressures equal and parallel to these, 
but in opposite directions, were applied at K, they would of 
themselves Bupport the mass, and the whole of the subjacent 
} TSB might be removed without affecting the equili- 
brium. (Art. 8.) [magine this to be done; call M the weight 
of the mass ASTD, and h the horizontal distance of its cen- 
tre of gravity from C, and letCK be represented by p, and 
the angle ECS by <\ then the perpendicular distances from 
pressures M + Y and P— X, imagined to be applied 
to Iv. are p Bin. 6 and p cos. d; therefore by the condition of 
the equality of moments, 



THE ANGLE OF KUPTURE LN THE ARCH. 



437 



(M+Y) P sin. d + (P-X) P cos.*=MA+Ya-Xy + Pp ; 



MA+Ys-Xy + Pp 

P ~(M+Y) sin.d + (P-X) cos. 6 ' 



. . (453), 




Q 



which is the equation to the line of resistance. 

M and h are given functions of 6 ; as also are X and Y, if 
the pressure of the load extend continuously over the surface 
of the extrados from D to T. 

It remains from this equation 
to determine the pressure P, be- 
ing that supplied by the opposite 
semi-arch. As the simplest case, 
let all the voussoirs of the arch 
be of the same depth, and let the 
\ J inclination ECP of the first joint 

\ \ i of the semi-arch to the vertical be 

\\| represented by 0, and the radii 

^ of the extrados and intrados by 

R and r. Then, by the known 
principles of statics.* 

B 6 

Mh=f /V sin. Wdr=-i(U 5 -r 5 )(cos. d-cos. 0) ; 

r 

also, M=i(R 3 -r a )(d-0); 

... p j£(K 2 -^)(6>-0) s i n . 6+Y sin. d-X cos. d + P cos. 6} = 

i(E 3 -r 3 )(cos.0-cos.^) + Yaj-Xy + Pp (454), 

which is the general equation to the line of resistance. 

The Angle of Rupture. 

337. At the points of rupture the line of resistance meets 
the intrados, so that there ?=r : if then Y be the correspond- 
ing value of 0, 

r {%(&*— r*)^— 0) sin. Y+Ysin. *— X cos. ^ + P cos. Y\ = 
i(R 3 -r s )(cos. e-cosy) + Yx-Xy + Tp (455). 



* See Note 1 at end of Part IV.— Ed. 



488 1HE an. ,i.i. of rupture 

at the points of rapture the line of resistance touche* 

the intrados, bo that there — = — =0 ; assuming then, tu 

simplify the results, that the pressure of the load is wholly 
in a vertical direction, so that X=0, and that it is colle 

over a Bingle point of the extrados, so that -jt=0, and dif- 

ferentiatiDg equation (454), and assuming -4<=0, when £=¥ 
and p=r, we obtain 

r jj i-_ / - (( ^_0) C os.Y + }(E 1 -r , )sin.T + 
V a. r— PauLTj^KR*— O sin - T j 
hence, assuming R=r(l+«), 

!> + a'(2a + 3) |tan.T= | ^_3a(a + 2)0} + 

3a (a + 2/F (456). 

Eliminating ('*'—©) between equations (455) and (456), we 
have 

| l> . + «'(** + *) | sec. a ^- | Yg + F -^ + a(K + a + l)C0S.Q j 

Bec.Y=— a(|*+l) (457). 

Eliminating P between equations (455) and (456), and 

reducing. 

Y* (j -.?sin.*+' ) ,, , w ^? _ N 

£(K + i«0 8in - Y — K a + ftl +i a O cos - -(i a, + a)cos.Y|sin.H r 

(±58). 



• This equation might have been obtained by differentiating equation (15-t) 

jp 

i 0, and assuming - - = when r and * are substituted for 
dd 
p and 0; for if that equation be represented by w=0, u being a function of 
., . a dn dV . du , <fw djo rfw ^ _, , efa „ . 

P - ' mi "' jp S + * ="' and * le + S=°- The same result Se =0 u 

therefore obtained, whether we assume — =0, or — =0, which last supposi- 

dd dl 

tion is that made in equation (456), whence equation (45S) has resulted. The 



IN THE AUCH. 



439 



Q 




Let AP = Xr; therefore ^= 



(1 + X) cos. ®. 

value of — , 
r 



Substituting this 



p|-sin.T + (l+X) C OS.ecos.^-l \ =(-|a 9 +a) 
il— (l + X)cos.©cos.^}(Y— ©) + (cos.^— cos.©)sin.Y i + 



Kia 2 +ia 3 )sin. ¥cos.0 



(459), 



by which equation the angle of rupture Y is determined. 

If the arch be a continuous segment the joint AD is ver- 
tically above the centre, and CD coinciding with CE, 0=0; 
if it be a broken segment, as in the Gothic arch, has a 
given value determined by the character of the arch. In the 
pure or equilateral Gothic arch, = 30°. Assuming 0=0, 
and reducing, 

Y ix 



(tan.— — X cot. v\ |=(ia a + a)| (tan.— — X cot. v\ 
¥-vers.¥ 1 +X(i a 2 +ia 9 ) (460.) 

It may easily be shown that as Y increases in this equa- 
tion, Y increases, and conversely ; so that as the load is 
increased, the points of rupture descend. When Y=0, or 
there is no load upon the extrados, 

. . (461). 



(tan. ^— X cot. yW— vers. Y + £X a _t^:=0 



hypotheses— =0, p 



r, determine the minimum of the pressures P, which 



being applied to a given point of the key-stone will prevent the semi-arch from 
turning on any of the successive joints of its roussoirs. 



440 



THE LINE 01 EE8I8TANCE. 



When «=0, or the load is placed on the crown of the 

arch, 

I = (i»V«)ver 8 .T-X(K+K) _ (Ja , +B)T my 

r tan.— — X cot. Y 

When tan. tt — ^ cot. Y 1 = 0,— becomes infinite ; 

/• \ z J t 

an infinite load is therefore required to give that value to the 

angle of rupture which is determined by this equation. 

Y . 
Solved in respect to tan. — , it gives, 



tan. 



t S+zQ'+^+s 



2 + X 



. (463). 



No loading placed upon the arch can cause the angle of rup- 
ture to exceed that determined by this equation. 



The line of resistance in a circular arch whose 
toub8oib8 are equal, and whose load is distrlbuted 
oyeb different points of lts extrados. 



. Let it be supposed that the pressure of the load is 
wholly vertical, and such that any 
portion FT of the extrados sustains 
the weight of a mass GFTV imme- 
diately superincumbent to it, and 
bounded by the straight line GY 
inclined to the horizon at the an- 
gle i\ let, moreover, the weight 
of each cubical unit of the load be 
equal to that of the same unit of 
the material of the arch, multiplied 
by the constant factor p ; then, re- 
presenting AD by K<3, ACF by ©, 
ACT by 0, and I)Z by 2, we have, 




area GFTV =/" TV 
8 



dz: 



SEGMENTAL ARCH. 



441 



but TY= MZ-(MT+ YZ), and MZ = CD = K-fKfr MT= 

R cos. 0, VZ = DZ tan. i = R sin. tan. i. Therefore MT+ 
YZ = R cos. 0-f-R sin. tan. i = R {cos. cos. 4 + sin. 6 sin. i] 
sec. 4 = R cos. {6—i) sec. 4 ; 

.% TY=R{l + /3-cos. (0-4) sec. «| ; 
also, s=DZ=R sin. 0; 

e e 

/. areaGFTY= /*TY . ^0=R a f{l+P- 


cos. (0—4) sec. i] cos. 0c?0; 

/. Y=weight of mass GFTY=rffc 2 f{l + (3-sec.t cos.{d-i)\ 


cos. Odd = fxR 2 1 (1 + /3) (sin. 0— sin. ©)— J sec. i {sin. (2 0— *) — 



sin. (2 0-4)}— J(0-0) ... (464).* 
e 
Yx = moment of GFTY= f^R 3 f \{1 + /3)— sec. « cos. (0- 4J 


sin. cos. 0^0=fxR s {i( 1 + / 3 )(cos. 2 0-cos. 2 0)-i(cos. 9 - 

cos. 3 0)— i tan. i (sin. s 0-sin. S ©)J . . . 465).* 



A SEGMENTAL ARCH WHOSE EXTRADOS IS HORIZONTAL. 

n 339. As the simplest case, let ns first 
suppose DY horizontal, the material of 
a the loading similar to that of the arch, 
b. and the crown of the arch at A, so 
that 4=0, f*'=l, and = 0. Substi- 
tuting the values of Y and Yx (equa- 
tions 464, 465) which result from tneso 
suppositions, in equation (455), solving 

that equation in respect to _, and re- 

p 

ducing, we have, — = 

See Note 2, at end of Part IV. — Ed. 





-i — n— - — i — ; — r— V 


I 


i '. i i i i i \ 


1 


I Is 1 I 1 1 1 


t : 


' : '. 1 1 1 i : 




li.. I l-V-pp- 




1 ' ! ' <\\\\ '■ 


i 


n r\\ \ LLUji 






Wc\\v^ r 


1 i 


_j 






i 


i 


KS^* i 






■| 








1 


V 


\ 




\ 


, 


\ 


, * 


\ ; 




\ ; 


' 


\! 



442 Tin: GOTHIC LRi II. 

♦ (l-Q)(14- Q )Vl 4-^)8in.^+i(l+a)'(l-2a)cos.'^4-(U , '+V-i)co3.^-^8in.t -fi 

l+A_cos.* (466). 



Assuming -=— =0 (see note, page 438.), and X = a, and 
reducing, 

|(1 - 2«) cos. *F - j(l_ a )(l+ /3) + (l+a)(l- 2a)}c0S. * + 
j (r | a)3 + 2 ( l- a Ml + /5)[co,^ ( ^il-(l + a) 

Iii the case in which the line of resistance passes through 
the bottom of the key-stone, so that X=0, equation (466) 
becomes 

I ^(l +a )XH-/3)(l-a)(l + COS.^)-Kl + a) a (l-2 a ) 

(1 + cos. Y) cos. Y—jYcot.jY 4-^=0 (466); 

whence assuming _;_=0, we have 

|(l+a) a (l-2a)cos.^-r(l + a) 2 {(l-a)/3+i(4-5a)f COS. Y + 
J!L - j(l + ay(l-a)(l+^) + l + a 2 (l+|a)}=0. . . .(469.) 



A GOTHIC ARCH, TOE EXTRADOS OF EACH SEMI-ARCH BEING 
A STRAIGHT LINE INCLINED AT ANY GIVEN ANGLE TO THE 
HoKIZoN, AND THE MATERIAL OF THE LOADING DIFFERENT 
I BOM I HAT OF THE ARCH. 

840. Proceeding in respect to this general case of the 
stability of the circular arch, by precisely the same steps as 
in the preceding simpler case, we obtain from equation 

Yx Y 

p < ia'+a'+a) (cos.9— cos.*)— (ia*+a)(*—e)sin.i4 — r — a sin.¥ 

^ = |cos.*-(l+A)co8.e} ,,,(470) 



THE GOTHIC AECH. 443 






in which equation the values of Y and Yx are those deter- 
mined by substituting Y for 6 in equations (464) and (465). 

Differentiating it in respect to Y, assuming-^— =0 (note, 
p. 438.), and X=a, we obtain 

(a + ia 2 — ia 3 — ^a 4 ) COS. sin. T — (£ a 2 + a ) sin. Y COS. ^ r — 
(ict 2 + a){l — (1+a) COS. COS. V\ (¥ — 0) 

Y Yx 
— {1 — (14- a) cos. Y cos. 0}~H — gsin. Y+ {cos. Y— 

/. x , J 1 ^( Y ^) sin. Y ^Y) ^ ,,_ 

(1 + a) COS. 6) { - 3 ^-_ -,--^0 . . . (471). 

Y Ya? 

Substituting in this equation the values of — 2 and — , de 

termined by equations (464) and (465) the following eqna 
tion will be obtained after a laborious reduction : it deter 
mines the value of Y : 

A + B cos. T— C cos. 2 Y + D cos. 3 *r+E sin. T— 
F sin. •*" cos. T — G sin. 3 Y— H cot. v + 

1(1 -K cos. T)t=^-|-t— ==0 (472) 

v ; em. "¥ sin. v v ' 

where 

A=Kl+a) 2 | |(l + a) tan i sin 3 0-(l+/3) J2-(l+a) cos" ©J 

-|(1+ a) COS. 3 1 +(2a + a 2 -a 3 — fa") COS. 

B = (l+a) 2 j2Kl-a 2 ) (1+/3) COS. 0-(l-f*)| +1. 
C = Kl+«) 2 {(l-«) (l + 0) + (l+a) (l-2a) COS. 0}. 
D=Ml+«) 9 (l-2a). 
E=,a(l+ a ) 2 (l— 2a) tan. t=|D tan. i. 
F=Kl+a) 3 (l-2a) tan. £ COS. ©=E(l+a) COS. 0. 
G=-|^(l+a) a (l— 2a) tan. £=D tan. i. 
H=£Kl+ a ) 3 |2(l + /3)- se c. « cos. (©-*)} sin. 20. 
I=l-(l-^)(l + a) 2 . 

K = (l-fa) COS. 0. 



L=(x(l+ a ) 2 J2(l-h/3)— sec. i cos. (0— t)}sin. 0. 
Tables might readily be constructed from thi 



s or any of 



4 1 1 an \i:< n SUSTAINING THE PRESSURE OF WATER. 

the preceding equations by assuming a »f values off, 

and calculating the corresponding values of 3 for each given 
value of a, '. i* 3 ' '• The tabulated results of such a series of 
calculations would Bhow the values of '+' corresponding to 

•i values of a, 3, '. I*, '- 1 - These values of ¥ being sub- 
Btituted in equation (470), the corresponding values of the 
horizontal thrust would be determined, and thence the polar 
equation to the line of resistance (equation 454:). 



A t IliCULAR ARCH HAVING EQUAL V0US30IRS AND SUSTAINING 
THE PRESSURE OF WATER. 

341. Let us next take a case of oblique pressure on the 
■v D extrados, and let us suppose it to be 

the pressure of water , whose surface 
stands at a height /3R above the sum- 
mit of the key-stone. The pressure of 
this water being perpendicular to the 
extrados will everywhere have its di- 
rection through the centre C, so that its 
motion about that point will vanish, 
and Y»— Xy=0; moreover, by the 
principles of hydrostatics,* the vertical 
component Y of the pressure of the water, superincumbent 
to the portion AT of the extrados, will equal the weight of 
that mass of water, and will be represented by the formula 
. it' we assume t=0. The horizontal component Xf of 
the pressure of this mass of water is represented by the 
formula 



X = arv a r;l + 3_cosJ}sin.^=Kl + a)V|(l4-/3)(cos.0- 
e 

5. *)— J (cos. a 0— cos. *$)\ (473). 

Resuming then 0=0, we have (equation 464), in respect 
to that portion of the extrados which lies between the crown 
and the points of rupture, 

p=Kl+*)M(l+0) ^n. Y-j sin. 2 *_£¥}, 

and (equation 473) ^-=^(1 + «) 9 {(1 + /3) vers. Y— i sin. *v\> 

• Sec Hydrostatics and Hydrodynamics, p. 30, 31. 

f See Note 3, end of Part IV.— Ed. 



AN ARCH SUSTAINING THE PRESSURE OF WATER. 445 

Y X 

.\- sin. Y_- cos. Y-a(l + a ) 2 |(l+/3) vers. Y-JY sin. Y} 

r T (474). 

Substituting this value in equation (455), making Yx— X.y=0, 
solving that equation in respect to — and making — =:1+X, 
we have 

P _ jk 2 +«-Ml+")1 *flin.*- j g +^+V-^(l+a)^ (l-f-/j)j vera. * 
r 2 A-f-vers. * 

If, instead of supposing the pressure of the water to be 

borne by the extrados, we suppose it to 

^^XXJX take effect upon the intrados, tending to 

/SQ^jSjS hlow up the arch, and if (3 represent the 

/C^MMSB height of the water above the crown of 

FT 1 S j fl^llllllil the intrados, we shall obtain precisely 

i | i ff |jlJgE|j||l the same expressions for X and Y as 

^nitepr; - -~3^y^ before, except that r must be substituted 

I m,/0W/M/sSWdm , £ or m Jj_ a y an( j ^ anc i y must be taken 

Y X 

negatively / in this case, therefore, — sin. Y— -j- cos. ^= 

— ^j(l + /3) vers. ¥— J*" sin. Yj ; whence, by substitution in 
equation (455), and reduction, 

F (ia 2 + a+ ^)YsinT- ja + * 2 + |a 3 +Kl+/3)}vers.Y ,_ 
r'~ X + vers.Y '^ } 

Now by note, page 438, , =0 ; differentiating equa- 
tions (475) and (476), therefore, and reducing, we have, 

Y J tan. -^— Xcot.^ | —vers. Y + AX=0 (477); 

which equation applies to both the cases of the pressure of a 
fluid upon an arch with equal voussoirs ; that in which its 
pressure is borne by the extrados, and that in which it is 
borne by the intrados; the constant A representing in the 



nrst case t lie quantity , ' = j~ — ^ — -, and m the 

second case; n „ = — . it the line ot resistance 

pass through the summit of the key-stone, X must be taken =a, 



146 EQUILIBRIUM OF AN ARCH. 

If it pass along the interior edge of the key-stone^ 

> -0. In tlii- Becond case, tan. — J^— sin. v\ = 0, therefore, 

r = o; bo that the point of rupture is at the crown of the 
arch. For this value of '■¥ equations (475) and (476) become 
vanishing fractions, whose values are determined by known 
methods of the differential calculus to be, when the pressure 
La od the extradoa, 

5=«_jrf+fl<l +»)*.... (478); 

when the pressure is on the intrados, 

J-._|rf_A» (479). 

It is evident that the line of resistance thus passes through 
the inferior edge of the key-stone, in that state of its equili- 
brium which precedes its rupture, by the ascent of its crown. 
The corresponding equation to the line of resistance is deter- 

P 

mined by substituting the above values of — in equation 

(454). In the case in which the pressure of the water is 
sustained by the intrados, we thus obtain, observing that 

X X 

-jsinJ— -j cos. 0=— f*,{(l+/3) vers. 6— £4 sin. 0J; 

a ' + o a _/^_(£ a s + a ' + a )cos. 6 

p - r (ia a + a+^)^sill.dH-( a -Ja 3 + M')C0sJ-Kl+^) * * * { } ' 

If for any value of 6 in this equation, less than the angle 
of the semi-arch, the corresponding value of p exceed 
(l+a)r, the line of resistance will intersect the extrados, and 
the arch will blow vp. 



Tin; EQUILIBRIUM OF AN ARCH, THE CONTACT OF WHOSE 
VOU8SOIR8 IS GEOMETRIC ALLY ACCURATE. 

34-2. The equations (459) and (456) completely determine 



EQUILIBRIUM OF AN ARCH. 



447 




Q 



the value of P, subject to the first 
of the two conditions stated in 
Art. 333., viz. that the line of re- 
sistance passing through a given 
point in the key-stone, determined 
by a given value of X, shall have a 
point of geometrical contact with 
the intrados. It remains now to 
determine it subject to the second 
condition, viz. that its point of ap- 
plication P on the key-stone shall 
be such as to give it the least va- 
lue which it can receive subject to 
the first condition. It is evident 
that, subject to this first condition, every different value of 
X will give a different value of "¥ ; and that of these values of 
¥ that which gives the least value of P, and which corres- 
ponds to a positive value of X not greater than a, will be the 
true angle of rupture, on the hypothesis of a mathematical 
adjustment of the surfaces of the voussoirs to one another. 
To determine this minimum value of P, in respect to the va- 
riation of Y dependent on the variation of X or of p, let it be 
observed that X does riot enter into equation (456) ; let that 
equation, therefore, be differentiated in respect to P and v, 

dP 

and let -j- be assumed =0, and Y constant, we shall thence 

CL i 

obtain the equation 






(481). 



whence, observing that 



i • «*, tan. ¥ 

ism. 2^= -== 

sec. ^ 



r ^ + a'(2tf + 3) 
3a(a + 2) 



tan. T. 



we obtain by elimination in equation (456) 

4Y 



sin. 2T— 2Y= 



-20 



(482), 



"a(a + 2y 

from which equation Y may be determined. Also by equa- 
tion (481) 



148 APPLICATIONS OF THB TIIKORY OF THE ARCH. 



!> ' '::-,(« + 2) cos. 9 ^-a 2 (2a + 3)S (483); 



and by eliminating sec. ^ between equations (457) and (481), 
and reducing, 

f=(l+X)C08. 3=p { /«(a + 2) j ^ + a 2 (ta + l) j - 

«(K + a + l)cos.e-i^l |. . . (484). 

The value of X given by this equation determines the actual 
direction of the line of resistance through the key-stone, on 
the hypothesis made, only in the case in which it is a positive 
quantity, and not greater than a; if it be negative, the line 
of resistance passes through the bottom of the key-stone, or 
if it be greater than a, it passes through the top. 

Such a mathematical adjustment of the surfaces of contact 
of the voussoirs as is supposed in this article is, in fact, sup- 
plied by the cement of an arch. It may therefore be con- 
sidered to involve 1 the theory of the cemented arch, the influ- 
ence on the conditions of its stability of the adhesion of its 
voussoirs to one another being neglected. In this settlement, 
an arch is liable to disruption in some of those directions in 
which this adhesion might be necessary to its stability. That 
old principle, then, which assigns to it such proportions as 
would cause it to stand firmly did no such adhesion exist, 
will always retain its authority with the judicious engineer. 

Applications of the theory of the arch. 

343. It will be observed that equation (459) or (472) 
determines the angle Y of rupture in terms of the load Y, 
and the horizontal distance x of its centre of gravity from 
the centre C of the arch, its radius r, and the depth a.r of its 
voussoirs ; moreover, that this determination is wholly inde- 
pendent of the angle of the arch, and is the same whether 
is arc be the half or the third of a circle; also, that if the 
angle of the semi-arch be less than that given by the above 
equation as the value of "*', there are no points of rupture, 
such as they have been defined, the line of resistance passing 
through the springing of the arch and cutting the intrados 



THEORY OF THE AECH. 449 

The value of "*" being known from this equation, P is 
determined from equation (456), and this value of P being 
substituted in equation (454), the line of resistance is com- 
pletely determined ; and assigning to d the value ACB 
(p. 437.), the corresponding value of p gives us the position 
of the point Q, where the line of resistance intersects the 
lowest voussoir of the arch, or the summit of the pier. 
Moreover, P is evidently equal to the horizontal thrust on 
the top of the pier, and the vertical pressure upon it is the 
weight of the arch and load: thus all the elements are 
known, which determine the conditions of the stability of a 
pier or buttress (Arts. 293. and 312.) of given dimensions 
sustaining the proposed arch and its loading. 

Every element of the theory of the arch and its abutments 
is involved, ultimately, in the solution in respect to Y of 
equation (459) or equation (472). Unfortunately this, solu- 
tion presents great analytical difficulties. In the failure of 
any direct means of solution, there are, however, various 
methods by which the numerical relation of y and Y may 
be arrived at indirectly. Among them, one of the simplest 
is this : — 

Let it be observed that that equation is readily soluble in 
respect to Y ; instead, then, of determining the value of Y 
for an assumed value of Y, determine conversely the value 
of Y for a series of assumed values of Y. Knowing the dis- 
tribution of the load Y, the values of x will be known in 
respect to these values of ¥, and thus the values of Y may 
be numerically determined, and may be tabulated. From 
such tables may be found, bv inspection, values of Y corres- 
ponding to given values of Y. 

The values of "*", P, and r are completely determined by 
equations (482, 483, 484), and all the circumstances of the 
equilibrium of the circular arch are thence known, on the 
hypothesis, there made, of a true mathematical adjustment 
of the surfaces of the voussoirs to one another ; and although 
this adjustment can have no existence in practice when 
the voussoirs are put together without cement, yet may it 
obtain in the cemented arch. The cement, by reason of 
its yielding qualities when fresh, is made to enter into so 
intimate a contact with the surfaces of the stones between 
which it is interposed that it takes, when dry, in respect 
to each joint (abstraction being made of its adhesive proper- 
ties), the character of an exceedingly thin ronssoir, having 
its surfaces mathematically adjusted to those of the adjacent 
voussoirs; so that if we imagine, not the adhesive properties 

29 



450 APPLICATIONS OF THE 






of the cement of an arch. but only those which tend to tho 
uniform diffusion of the sh its man 

r int<> the conditions of its equilibrium, these equations 
;he entire tL the cemented arch. The fa 

probably includes all that can be n 
upon in the properties of cement as applied to large struc- 

An arch may fall either by the sinking or the rising of 
its crown. In the farmer case, the line of resistance passing 
through the top of the kev-~t»ne is made to cut the extrados 
beneath the points of rupture; in the latter, passing through 
the bottom of the key-stone, it is made to cut the extr 
between the points of rupture and the crown. 

In the first case the values of X. Y, and P, being deter- 
mined as before and substituted in equation (454), and p 
a assumed = 1 —a/-, the value of \ which corresponds 
. will indicate the point at which the line of 
cuts the extrados. If this value of <? be less than 
the angle of the semi-arch, the intersection of the line of 
stance with the extrados will take place above the 
_ _. h will fall. 

En - 36, in which the crown ascends, let the 

.be determined from equation (454 9 p 
_ -lined =r ; if this value of p be greater than li. 
the corresponding value of <? less than the angle of rupture, 
iine oi resistance will cut the extrados, the arch will 
open at the intrados, and it will fall by the descent of the 
crown. 

If the load be collected over a single point of the arch, 
the intersection of the line of resistance with the extrados 
will take place between this point and the crown ; it is that 
portion only of the line of resistance which lies :hese 

points which enters therefore into the discussion. Xow if 
we i Art. 336.. it will be apparent that in respect to 

i the line, the values of X and Y in equations 
and (45 - i be neglected ; the only influence of 

these quantities being found in the value of P. 



THEOET OF THE AECH. 



451 




-Let a circular arch of equal voussoirs have 
the depth of each voussoir equal to 
T Vth the diameter of its intrados, so 
that a='2, and let the load rest upon 
it by three points A, B, D of its 
extrados, of which A is at the crown 
and B D are each distant from it 45° ; 
and let it be so distributed that f ths 
of it may rest upon each of the points 
B and D, and the remaining J upon 
A ; or let it be so distributed within 
60° on either side of the crown as to 
produce the same effect as though it 
rested upon these points. 

Then assigning one half of the load 
upon the crown to each semi-arch, 
and calling x the horizontal distance 
of the centre of gravity of the load upon either semi-arch 

rp 

from C, it may easily be calculated that - = f sin. 45 = 

•5303301. Hence it appears from equation (463) that no 
loading can cause the angle of rupture to exceed 65°. 
Assume it to equal 60°; the amount of the load necessary to 
produce this angle of rupture, when distributed as above, 
will then be determined by assuming in equation (460), 

^=60°, and substituting a for X, -2 for a, and -5303301 for?. 

Y Y 

We thus obtain -^-OlSS. Substituting this value of -*, and 

also the given values of a and v in equation (457), and 
observing that in this equation^ is to be taken =1+ a and 



0=0, we find —. = -11832. Substituting this value of -s in 

the equation (454), we have for the final equation to the line 
of resistance beneath the point B 



•2426 vers. ^ + -1493 



•0138 sin. 6 + -1183 cos. 6 + -22 6 sin. 6' 






APPLICATIONS OF THE 




If the arc of the arch be a com- 
plete semicircle, the value of p in this 

if 
equation corresponding to 6 = - will 

determine the point Q, where the 
line of resistance intersects the abut- 
ment ; this value is p=l-09r. 

If the arc of the arch be the third 

o of a circle, the value of p at the 

abutment is that corresponding to 

if 
•= -; this will be found to be r, as 

o 

it manifestly ought to be, since the 
points of rupture are in this case at 
the springing. 

In the first case the volume of the semi-arch and load is 
represented by the formula 

'•'{»«' + «) J + 5 [=-3594r' ; 
and in the second case by 

Thus, supposing the pier to be of the same material as the 
arch, the volume of its material, which would have a weight 
equal to the vertical pressure upon its summit, would in the 
first case be -3594/ >a , and in the second case -2442/' 3 , whilst 
the horizontal pressures P would in both cases be the same, 
viz. •11S32/- 5 ; substituting these values of the vertical and 
horizontal pressures on the summit of the pier, in equation 
(377), and for k writing -J a— (p— r), we have in the first 
case 



H= 



and in the second case, 
H= 



3594(g— *09r)r\ 

•11S3LV-K ' 



■2442 ar 



•llS32r a -ia a 



THEOKY OF THE AECH. 453 

where H is the greatest height to which a pier, whose width 
is a, can be built so as to support the arch. 

If |a 2 — -11832^=0, or a=-4864^, then in either case the 
pier may be built to any height whatever, without being 
overthrown. In this case the breadth of the pier will be 
nearly equal to £th of the span. 

The height of the pier being given (as is commonly the 
case), its breadth, so that the arch may just stand firmly 
upon it, may readily be determined. As an example, let us 
suppose the height of the pier to equal the radius of the 
arch. Solving the above equations in respect to a, we shall 
then obtain in the first case & = *2978r, and in the second 
a = 'Sr. 

If the span of each arch be the same, and r x and r 3 repre- 
sent their radii respectively, then r{=r a sin. 60* ; supposing 
then the height of the pier in the second arch to be the same 
as that in the first, viz. t\, then in the second equation we 
must write for H, r 2 sin. 60°. We shall thus obtain for a the 
value -28r 2 . 

The piers shown by the dark lines in the preceding 
figures are of such dimensions as just to be sufficient to 
sustain the arches which rest upon them, and their loads, 
both being of a height equal to the radius of the semicircular 
arch. It will be observed, that in both cases the load 
Y=;0138r 3 , being that which corresponds to the supposed 
angle of rupture 60°, is exceedingly small. 



Example 2.— Let us next take the example of a Gothic 
arch, and let us suppose, as in the last examples, that the 
angle of rupture is 60°, and that a='2; but let the load in 
this case be imagined to be collected wholly over the 

crown of the arch, so that - = sin. 30°. Substituting in equa- 

r 

tion (459), 30° for 0, and 60° for ¥, and *2 for a, and sin. 30° 

x . Y 

for -, we shall obtain the value -21015 for — : whence by 

p 

equation (457)—= -2405, and this value being substituted. 






API': 5 OF THE 



equation (454) gives l'145r for 

the value of p when 6 = _. We 

have thus all the data for deter- 
mining the width of a pie 
given height whieh will ju>t 
support the arch. Let the 
height of the pier be supj 
as before, to equal the radius 
of the intradoa : then, since the 
weight of the semi-arch and its 
load is '5556V, and the horizon- 
tal thrust -2405/ 2 , the width a 
of the pier is found bv equation 
(379) to be 4195r. 
The preceding figure represents this arch ; the square, 
farmed by dotted lines over the crown, shows the dimen- 
oi the load of the same materials as the arch which will cause 
the angle of the rupture to become 60° ; the piers are of the 
required width '4U*5/'. such that when their height is equal 
- shown in the n'giire, and the arch bears this insist- 
ent pre«ure, they may be on the point of overturning. 




Tables 



OF THE THRUST OF AHCHES. 



344. It is not possible, within the limits necessarily 
asigned to a work like this, to enter further upon the dis- 
- questions whose solution is involved in the 
.lions which have been given ; these can, after all, be- 
ae »le t<> the reader, only when tables shall 

nned from them. 
S ich tables have been calculated with great accuracy by 
ML I laridel in respect to that case of a segmental arch* w 
loadii g - the same material as the voussoirs, and the ex- 
■ i-arch a straight line inclined at any given 
angle to the horizon. These tables are printed in the Ap- 
pendix (Tab! - _ 



The term segmental arch is used, here and elsewhere, to distinguish th* 
"f the circular arch in which the intrados is a contiguous segment froi 
that in which it is composed of two segments struck from different centres, a 
in the Gothic arch. 



THEORY OF THE AECH. 455 

Adopting the theory of Coulomb*, M. Garidel has arrived 
iA an equationf which becomes identical with equation (472) 
in respect to that particular case of the more general condi- 
tions embraced by that equation, in which f*.=l and 0=0. 

By an ingenious method of approximation, for the details 
of which the reader is referred to his work, M. Garidel has 
determined the values of the angle of rupture Y, and the 

P 

quantity -r& in respect to a series of different values of a and 

(3. The results are contained in the tables which will be 
found at the end of this volume, 
p 
The value of -^ being known from the tables, and the 

values of Y and Yx from eouations (464), (465), the line of 
resistance is determined by "the substitution of these values 
in equation (454). The line of resistance determines the 
point of intersection of the resultant pressure with the sum- 
mit of pier ; the vertical and horizontal components of this 
resultant pressure are moreover known, the former being the 
weight of the semi-arch, and the other the horizontal thrust 
on the key. All the elements necessary to the determina- 
tion of the stability of the piers (Arts. 289 and 312) are 
therefore known. 

It will be observed that the amount of the horizontal 
thrust for each foot of the width of the soffit is determined 

p 
by multiplying the value of — 2 , shown by the tables, by the 

square of the radius of the intrados in feet, and by the 
weight of a cubic foot of the material 



* See Mr. Harm's Theory of Bridges, Art. 16. ; also p. 24. of the Memoir on 
the Arch by the author of this work, contained in the same volume, 
f Tables des Poussees des Voutes, p. 44. Paris, 1837. Bachelier. 



456 NOTES TO PART IY. 



Son 1. Taxt TV. 



The length of an elementary arc as of the intrados AS subtending the angle 
dfl is expressed by rdd ; an elementary Tolume of the arch will therefore be 
expressed t e perpendicular distance of the centre of gravity of 

this Tolume from the vertical line ---:.. 5; the moment of this volume, 

with regard to CE, is therefore rdddrXr em.d=r*dr sin. 0cV; then from (Art. 
there obtains 

R 6 

*M=ff*Jr I 

r e 

Xotx 2. Pjjit IT.— General integral* of equation* 464, 465. 

The general integral, (equation 464) 

I - -eoe,(0-*)sec i} co*ft#=/ \- cos. 0#- 

/ i :. t (cos. 6 cos. i-f-sm- ^ ^^ coa - 5rfr= / "-— - : cos. 5d5— 

/sec. i coe. t cofl. s 6d9 — / sec * an. t sin. 5 coa. &tf. 

But J 1— DO& ^^=(14-3) sin. 5; /sec. < cos. t coa. s 5d5= 

■ec t cos. i j (- -| ^-)<#=sec. t co *>'(a6+r sin ' — j*/ 86 ** * ■"* * 

/I 1 

-sin. 2 ri 2? = — sec, t sin. t - coi 

.J - -coa. (5-0 sec <£coa. d&={\+$) ain. 5-A sec i 
(sin. 25 coa. i— sin. i coa. 25)— -5=(l-f-3) sin. 5 sec. i sin. (25— i)—^ 

■i - 

The general integral, | - :. t cos. (5— t ){ an. 5 coa. 5d5, (equatioa 

/ .- =in.5cos. 6dB— f ;.5cos. i+an.5ain. i ! sin. 5 cos. 5*. 

But f (1+3) sin. 5 coa. ddt>=f{l+ 3) ™' **'* =-?p+3) coa. 25= 



NOTES TO PART IV. 457 



/ sec. t] cos. 0cos. j-4-sin. 0sin. i > sin. 6 cos. 6d . 8= /cos. 2 (9sin. ftc?0-j- 
/tan. i sin. 2 0cos. Odd . — /cos. 2 0<#cos. 0-4-/ tan - t sin. W sin. 0= 



cos. *d-\ — sin. 3 

3 ^3 



■f \ (!+ 5 ) ~ sec - « cos - (^-0 C sin - cos. 0d . -f-<$=_I(l-f-/3) cos. a 0+ 
j(14-/3)-f-^-cos. 3 0-^-tan. i sin. 3 0.tan t . 



Note 3. Part IV. 

In equation (42*7), (Art. 319), by making 0=0, we obtain P=£ fi x a? ; since 

tan. 7=1] and this answers to the case of .the horizontal pressure of a perfect 

fluid like water. From this expression there obtains dT=fj, 1 xdx, to express 
the elementary pressure at any depth x below the surface. This depth in 
(Art. 341), equation (4*73), is TV=AD-f AB=AD-f-AC-BC=/ffi-j-R-R cos.P, 

.-. tfP=uR(l-J-/3-cos. 0)R^(l-hS-cos. 0)=/*R 2 ji-f /?_ C os. 0} sin. ddd 

6 
.-. Y=X=[iR: 2 f jl+,3-cos. } sin. ddd. 



PAR T V. 

THE STRENGTH OF MATERIALS. 



ELASTICITY. 

345. From numerous experiments which have been made 
upon the elongation, flexure, and torsion of solid bodies 
under the action of given pressures, it appears that the 
displacement of their particles is subject to the following 
laws. 

1st. That when this displacement does not extend beyond 
a certain distance, each particle tends to return to the place 
which it before occupied in the mass, with a force exactly 
proportional to the distance through which it has been 
displaced. 

2dly. That if this displacement be carried beyond a 
certain distance, the particle remains passively in the new 
position which it has been made to take up, or passes finally 
into some other position different from that from which it 
was originally moved. 

The effect of the first of these laws, when exhibited in 
the joint tendency of the particles which compose any finite 
mass to return to any position in respect to the rest of the 
nia>>, or in respect to one another, from which they have 
been displaced, is called elasticity. There is every reason to 
believe that it exists in all bodies within the limits, more or 
less extensive, which are imposed by the second law stated 
above. 

The force with which each separate particle of a body 
tends to return to the position from which it has been 
displaced varying as the displacement, it follows that the 
force with which any aggregation of such particles, consti- 
tuting a finite portion of the body, when extended or 
compressed within the limits of elasticity, tends to recover 
rm, that is the force necessary to keep it extended or 



ELONGATION. 459 

compressed, is proportional to the amount of the extension 
or compression ; so that each equal increment of the extend- 
ing or compressing force produces an equal increment of its 
extension or compression. This law, which constitutes 
perfect elasticity, and which obtains in respect to fluid and 
gaseous bodies as well as solids, appears first to have been 
established by the direct experiments of S. Gravesande on 
the elongation of thin wires.* 

It is, however, by its influence on the conditions of 
deflexion and torsion that it is most easily recognized as 
characterizing the elasticity of matter, under all its solid 
forms,f within certain limits of the displacement of its 
particles or elements, called its elastic limits. 



Elongation. 

346. To determine the elongation or compression of a bar of 
a given section under a given strain. 

Let K be taken to represent the section of the bar in 
square inches, L its length in feet, I its elongation or com- 
pression in feet under a strain of P pounds, and E the strain 
or thrust in pounds which would be required to extend a 
bar of the same material to double its length, or to compress 

* For a description of the apparatus of S. Gravesande, see Illustrations of 
Mechanics, by the Author of this work, 2d edition, p. 30. In one of his 
experiments, Mr. Barlow subjected a bar of wrought iron, one square inch in 
section, to strains increasing successively from four to nine tons, and found the 
elongations corresponding to the successive additional strains, each of one ton, 
to be, in millionths of the whole length of the bar, 120, 110, 120, 120, 120. 
In a second experiment, made with a bar two square inches in section, under 
strains increasing from io tons to 30 tons, he found the additional elongations, 
produced by successive additional strains, each of two tons, to be, in millionths 
of the whole length, 110, 110, 110, 110, 100, 100, 100, 100, 95, 90. From an 
extensive series of similar results, obtained from iron of different qualities, he 
deduced the conclusion that a bar of iron of mean quality might be assumed 
to elongate by 100 millionth parts, or the 10,000th part, of its whole length, 
under every additional ton strain per square inch of its section. {Report to 
Directors of London and Birmingham Railway. Fellowes, 1S35.) 

The French engineers of the Pont des Invalides assigned 82 millionth parts 
to this elongation, their experiments having probably been made upon iron of 
inferior quality. M. Vieat has assigned 91 millionth parts to the elongation 
of cables of iron wire (Xo. 18.) under the same circumstances, MM. Minard 
and Desormes, 1,17(5 millionth parts to the elongation of bars of oak. 
(lllust. Mcch. y p. 393.) 

f The experiments of Prof. Robison on torsion show the existence of this 
property in substances where it might little be expected; in pipe-clay, for 
instance. 



4G0 THE KOBE EXPENDED ON ELONGATION. 

it to one half its length, if the elastic limit of the material 
were such as to allow it to be so far elongated or compressed, 
the law of elasticity remaining the same.* 

Now, Buppose the bar. whose section is K square inches, 
to be made up of others of the same length L, each one inch 
in section; these will evidently be K in number, and the 

P 
strain or the thrust upon each will be represented by ^. 

Moreover, each bar will be elongated or compressed, by this 
strain or thrust, by I feet; so that each foot of the length of 
it (being elongated or compressed by the same quantity as 
each other foot of its length) will be elongated or compressed 

by a quantity represented, in feet, by y. But to elongate 

or compress a foot of the length of one of these bars, by one 
foot, requires (by supposition) E pounds strain or thrust ; to 

elongate or compress it by j- feet requires, therefore, E^ 
pounds. But the strain or thrust which actually produces 
this elongation is tf pounds. Therefore,^ = E^. 
PL 



34:7. To find the number of units of work expended upon the 
elongation by a given quantity (I) of a bar whose section is 
K and its length L. 

If x represent any elongation of the bar (x being a part 
of Q, then is the strain P corresponding to that elongation 

KE 

represented (equation 485) by -y-#; therefore the work 

done in elongating the bar through the small additional 

KE 

space -Xf, is represented by -^—x±x (considering the strain 

to remain the same through the small space &x) ; and the 

* The value of E in respect to any material is called the modulus of its elas- 
ticity. The value of the moduli of elasticity of the principal materials of con- 
struction hare been determined by experiment, and will be found in a table at 
the end of the volume. 



THE WORK EXPENDED ON ELONGATION. 461 

whole work U done is, on this supposition, represented by 

-KE 

-j-lx&x, or (supposing Ax to be infinitely small) by 

KE / 7 , n KE 72 
-j— J %dx or by \rr~ "• 

.•.U=W- (486). 



KE 
348. By equation (485) Y =^-l, therefore U^-JPZ; 

whence it follows that the work of elongating the bar is one 
half that which would have been required to elongate it by 
the same quantity, if the resistance opposed to its elongation 
had been, throughout, the same as its extreme elongation I. 
If, therefore, the whole strain P corresponding to the 
elongation I had been put on at once, then, when the elonga- 
tion I had been attained, twice as much work would have 
been done upon the bar as had been expended upon its 
elasticity. This work would therefore have been accumu- 
lated in the bar, and in the body producing the strain under 
which it yields ; and if both had been free to move on (as, 
for instance, when the strain of the bar is produced by a 
weight suspended freely from its extremity), then would 
this accumulated work have been just sufficient yet further 
to elongate the bar by the same distance I* which whole 
elongation of 21 could not have remained ; because the 
strain upon the bar is only that necessary to keep it 
elongated by I. The extremity of the bar would therefore, 
under these circumstances, have oscillated on either side of 
that point which corresponds to the elongation I. 

* The mechanical principle involved in this result has numerous applica- 
tions; one of these is to the effect of a sudden variation of the pressure on a 
mercurial column. The pressure of such a column varying directly with its 
elevation or depression, follows the same law as the elasticity of a bar; 
whence it follows that if any pressure be thrown at once or instantaneously 
upon the surface of the mercury, the variation of the height of the column 
will be twice that which it would receive from an equal pressure gradually 
accumulated. Some singular errors appear to have resulted from a neglect of 
this principle in the discussion of experiments upon the pressure of steam, 
made with the mercurial column. No such pressure can of course lie made to 
operate, in the mathematical sense of the term, instantaneously; and the term 
gradually has a relative meaning. All that is nieani is, that a certain relation 
must obtain between the rate of the increase of the pressure ami the amplitude 
of the motion, so that when the pressure no longer increases the motion may 
cease. 



}.;-J BJ -III! \. i: AND FRAGLITT. 

849. Eliminating I between equations (485) and (486), we 
obtain 

tf=*l3 (487); 

whence it appears that the work expended upon the elonga- 
tion of a bar under any strain varies directly as the square 
of the Btrain and the Length of the bar, and inversely as the 
area of its section.* 



The Moduli of resilience and fragility. 

350. Since U=iE(-jJ KL (equation 486), it is evident 

that the different amounts of work which must be done upon 
different bare of the same material to elongate them by equal 

fractional parts I y), are to one another as the product KL. 

Let now two such bars be supposed to have sustained that 
fractional elongation which corresponds to their elastic limit • 
Let U« represent the work which must have been done upon 
the one to bring it to this elongation, and M that upon the 
other : and let the section of the latter bar be one square 
inch and its length one foot ; then evidently 

U«=MJKX (488). 

M« is in this case called the modulus of longitudinal resili- 
ence, f 

It is evidently a measure of that resistance which the 
material of the bar opposes to a strain in the nature of an 
impact^ tending to elongate it beyond its elastic limits. 

If M/be taken to represent the work which must be simi- 
larly done upon a bar one foot long and one square inch in 
section to produce fracture, it will be a measure of that 
resistance which the bar opposes to fracture under the like 
circumstances, and which resistance is opposed to its fra- 

* From this formula may be determined the amount of work expended pre- 
judicially upon the elasticity of rods used for transmitting work in machinery, 
under a reciprocating motion — pump rods, lor instance. Xsvdden effort of the 
pressure transmitted in the nature of an impact may make the loss of work 
doable that represented by the formula ; the one limit being the minimum, and 
the other the maximum, of the possible loss. 

f The term "modulus of resilience" appears first to have been used by 
Mr. Tredgold in his work on "the Strength of Cast Iron," Art. 3U4, 



A BAR SUSPENDED VERTICALLY. 463 

gility ; it may therefore be distinguished from the last men- 
tioned as the modulus of fragility. If U/ represent the work 
which must be done upon a bar whose section is X square 
inches and its length L feet to produce fracture; then, as 
before, 

U / =M / EX (489). 

If T e and P/ represent respectively the strains which 
would elongate a bar, whose length is L feet and section K 
inches, to its elastic limits and to rupture ; then, equation 
(487), 

U=M..KL=i||; 

•' Me=i m- Similarly M '=*i8 (490 >- 

These equations serve to determine the values of the 
moduli M„ and M/by experiment.* 



351. The elongation of a bar suspended vertically, and sus- 
taining a given strain in the direction of its length, the 
influence of its own weight being taken into the account. 

Let x represent any length of the bar before its elonga- 
tion, Ax an element of that length, L the whole length of the 
bar before elongation, w the weight of each foot of its 
length, and K its section. Also let the length x have become 
x x when the bar is elongated, under the strain P and its own 
weight. The length of the bar, below the point whose dis- 
tance from the point of suspension was x before the elonga- 
tion, having then been L— x, and the weight of that portion 
of the bar remaining unchanged by its elongation, it is still 
represented by (L— x) w. Now this weight, increased by P, 
constitutes the strain upon the element Ax\ its elongation 
under this strain is therefore represented (equation 485) by 

A~=j — - — Ax, and the length Ax l of the element when thus 

* The experiments required to this determination, in respect to the princi- 
pal materials of construction, have been made, and are to be found in the 
published papers of Mr. tfodgkinson and Mr. Barlow. A table of the moduli 
of resilience and fragility, collected from these valuable data, is a desideratuic 
in practical science. 



•I'll TB OF 

elongated, by &x-i YF ^ whence dividing by &x. 

and pa i the limit, we obtain 

Integrating between the limits and L, and representing 
by L, 3ie Length of* the elongated rod, 

H 1+ t&) l+ ^ l '* (492) - 

If the strain be converted into a thrust^ P must be made 
to asBnme the negative sign; and if this thrust equal one 
half the weight of tbe bar, there will be no elongation at all. 



352. The vertical oscillations of ax elastic eod or 
cord sustaining a given weight suspended from rts 
extremity. 

Let A represent the point of suspension of the rod (fig. 1. 
on the next page), L its length AB before its elongation, and 
V the elongation produced in it by a given weight A\ T sus- 
pended from its extremity, and C the corresponding position 
of the extremity of the rod. 

Let the rod be conceived to be elongated through an 

additional distance CD=<? by the application of any other 

given strain, and then allowed to oscillate freely, carrying 

with it the weight A\ r ; and let P be any position of its 

mity during any one of the oscillations which it will 

thus be made to perform. If, then, CP be represented by a, 

the corresponding ejongation BP of the rod will be repre- 

Bented by i/~-,'.\ and the strain which would retain it perma- 

KE 
nently at this elongation (equation 485) by -j-(^+»); the 

unbalanced pressure or moving force (Art. 92.) upon the 
period of this elongation, will therefore be 

represented by —-($+ x )—W) or by -p^; since W, being 

the Btrain which would retain the rod at the elongation £Z, is 

lv E 
represented by -— \l (equation 4S5). 

* WhewelTs Analytical Statics, p. 113. 



A LOADED BAR. 



465 



The unbalanced pressure, or moving force, upon the mass 
W varies, therefore, as the distance x of the point P from the 
given point C ; whence it follows by the general principle 
established in Art 97., that the oscillations of the point P 
extend to equal distances on either side of the point C, as a 
centre, and are performed isochronously, the time T of each 
oscillation being represented by the formula 



\gEE/ 



(493). 



The distance from A of the centre C, about which the 
oscillations of the point P take place, is represented by 
L-f ££; so that, representing this distance by L l5 and substi- 
tuting for \l its value, we have 



A 



Ad 



A 



L + 



WL 
KE 



. (494). 



353. Let us now suppose that when in 
making its first oscillation about 
(Jig. 2.) the weight "W" has attained its 
highest position 2 n and is therefore, for 
an instant, at rest in that position, a 
second weight w is added to it ; a second 
series of oscillations will then be com- 
menced about a new centre C^ whose 
distance L 2 from A is evidently repre- 
sented by the formula 

L, = I+(Z>^....(495). 



T 

So that the distance QQ 1 of the two centres is — — ; and the 

greatest distance C^, beneath the centre C,, attained in the 
second oscillation, equal to the distance, C/Z, at which the 
oscillation commenced above that point. Now G 1 D 1 =s 

C 1 d. = 07 l + CC 1 =CD + CC 1 =c+ ^; the amplitude ^D, of 



the second oscillation is therefore 2f 

30 



c + 



KE 



4^6 -nn-: oscillations of a loaded bar. 

Tot the w.-hd.t '"• he conceived to be removed when the 
. ,,„;,„ ir„f the second oscillation is attained, a. third 
. ,-, filiations wil] .hen he commenced tile position of 
,., centre being determined by equation (494), is identical 
1 h of the lentre C, about which the first oscillation 
• ' , erformed. I" its third oscillation the extremity of the 
, SfflK» ascend to a point i as far above the point 
C h 1) is below it; so that the amplitud e of this th ird oscil- 
lation is represented by 2CD„ or by 20 1 D 1 + CC„ or by 
2 ( e -|- 2wL ) When the highest point <Z, of this third oscil- 

htion is attained, let the weight w be again added ; a fourth 
oscilhition will then be commenced, the position of whose 
eent e wnl be determined by equation (495,) and will there- 
foe be identical with the centre C about which the second 
oscillation was performed ; so that the greatest distance ,C,D, 
beneatbhat point attained in this fourth oscillation will be 
equ-il to Cy! or to CO, 4- CD,; and its amplitude will be 

represented by 2 (c+*g). And if the weight v, be thus 

conceived to be added continually, when the highest point 
of each oscillation is attained, and taken off at the lowest 
point it is evident that the amplitudes of these oscillations 
ilu continually increase in an arithmetics .1 series ; so 
Sat the amplitude "A. of the # oscillation will be repre- 
sented by the formula 

A.= 2Jo + («-l)g}----(^). 

The ascending oscillations of the series being made about 
the centre C, and the descending oscillations about , 
if „ be an even number, the centre of the «<" osclla ion ^ 
C, ; the elongation c n of the rod corresponding^ the lowest 
Mint of this oscillation is therefore equal to BO.+jA,; or 
Bubstltnting for BC, its value given by equation (495), and 
for A. its value from equation (496), 

r> _,l (W + «»)L (497) . 

Thus it is apparent that by the long continued and 



DEFLEXION. 



467 



penod>ca addition and subtraction of a weight w, so small 

rodwC r, bU L a i ,ight Ration or contraction of the 
rod when first added or removed from it, an elongation 
may eventually be produced, so great as to pass Ss of it^ 

verified^' e 7 6 ? t0 t , bre ^ {t ^ UmerOUS observations have 
venfied this fact: the chains of suspension bridges have 

S.™ X Z bjt \ 6 m , easured tread of soldiers;* and M 
Savart has shown, that by fixing an elastic rod at its centre' 

vals^r/wth?"^- fiD < al -g » -t measured "! ; ! 
vats, it may, by the strain resulting from the slio-ht friction 
received thus periodically upon ifs surface, be made w^th 
great ease to receive an oscillatory movement of sufficTeut 
amplitude to be measured^ M. Poncelet has compared the 
measurement of M. Savart with theoretical deductions 
^%S£?jL ? *»** •*-* and has sh'owii 

Deflexion. 
354. The neutral surface of a deflected learn. 
One surface of a beam becoming, when deflected convex 

iW W 0t m'' C 7T V* i8 eyiden * that Ae materiaTS 
ng that side of the beam which is bounded by the one 
surface is, m the act of flexure, extended, and that of the 
other compressed. The surface which separates these two 
portions of the material being that whereas extension ter- 
minates and its compression begins, and which susta n 

™r'sCl h oI eXtenSi ° n n ° r C0 ^~'- calledTc 

355. The position of the neutral surface of a beam. 
Let ABCD be taken to represent any thin lamina of the 

SKte Pa" s mT.hTe ,t e f ; he r «rsr*s •»">««» ^ u. rZ 

a q,,q„«« • u i )y that the durat »on of the oscil at ons of the chain* nf 

» S Y:t,r y '" Certain r 6es «-d to nearly six seonds; there 

any number' 'fi,? ft' *"** "'!" i3 o« h ™"'^ «t each in,,,-,;,!, or after 

LosdlSlr,,. ,1 r V j S ' be , tWCeD " ,e raarehin g "ep of the troops and the 

f Mccanique Indmtrielle, p. 437, Art. 331.— Ed. 



40* 



THE NEUTRAL SURFACE 




beam contained by planeg 
parallel to the plane of its 
deflexion, and P„ P a , P, the 
resultants of all the pres- 
sure- applied to it ; ach that 
portion of the neutral sur- 
face of the beam which is 
contained within this la- 
mina, and may be called its 
neutral line; PT and QV 
planes exceedingly near to 
one another, and perpen- 
dicular to the neutral line at the points where they intersect 
it ; and O the intersection of PT and QV when produced. 

Now let it be observed that the portion ArTD of the 
beam is held in equilibrium by the resultant pressure P„ 
and by the elastic forces called into operation upon the sur- 
face PT ; of which elastic forces those acting in PR (where 
the material of the beam is extended) tend to bring the 
points to which they are severally applied nearer to the 
plane SQ, and those acting in PT (where the material is 
compressed), to carry their several points of application 
farther from the plane SY. 

Let aR=x, SR=a#, and imagine the lamina PQYT to be 
made up of fibres parallel to SR ; then will a# represent 
the length of each of these fibres before the deflexion of the 
beam, since the length of the neutral fibre SR has remained 
unaltered by the deflexion. Let dx represent the quantity 
by which the fibre pq has been elongated by the deflexion 
of the beam, then is the actual length of that fibre repre- 
sented by &x-\-dx. Whence it follows (equation 485), that 
the pressure which must have operated to produce this 

6x 
elongation is represented by E — A&, Ajc being taken to repre- 

sent the section of the fibre, or an exceedingly small element 
of the section PT of the lamina. Now PL and QY being 
normals to SR, the point O in which they meet, when 
] >rod need, is the centre of curvature to the neutral line in 
U. Let the radius of curvature OR be represented by R, 

and the distance Up by p. By similar triangles, tt4 = 



SR 



or 



R + P_A#-f 6x 
R " ±x 



, or 1 + 



R 



6x 



OR 

p 

therefore, -jy = 



OF A BEAM. 469 

til* OT* 

— . Substituting this value of — in the expression for the 

pressure which must have operated to produce the elonga- 
tion of the fibre pq, and representing that pressure by aP, 
we have 

aV=-^aJc (498). 

If, therefore, EP be represented by lc l and KT by & 2 , then 
the sum of the elastic forces developed by the extension of 

the fibres inRPQS is represented by^-2 pA/fc; and, similarly, 

the sum of those developed bj the compression of the fibres 

E * 2 
in RTVS is represented by =-2 pA&. Now let it be observed 

that (since the pressures applied to APTD, and in equili- 
brium, are the forces of extension and compression acting 
in HP and RT respectively, and the pressure Pj), if the 
pressure P x be resolved in a direction perpendicular to the 
plane PT, or parallel to the tangent to the neutral line in R, 
this resolved pressure will be equal (Art. 16.) to the differ- 
ence of the sums of the forces of extension and compression 
applied (in directions perpendicular to that plane, but oppo- 
site to one anofher) to the portions RP and RT of it respec- 
tively. Representing, therefore, by d the inclination R^P, 
of the direction of P, to the normal to the neutral line in R, 
we have 

E *» E * 2 

P 1 sin. d=-2 op A&-_2 pA&. 

But if Jc be taken to represent the whole section PT, and h 
the distance of the point R from its centre of gravity, then 
(Art. 18.) 

JK 

RP 
;.h=-^&mJ (499); 

which expression represents the distance of the neutral line 
from the centre of gravity of any section PT of the lamina, 
that distance being measured towards the extended or the 
compressed side of the lamina according as d is positive or 



470 RADIUS OF CURVATURE. 

negative; BO that the neutral line passes from one side to 
the other of the line joining the centres of gravity of the 
ions of the lamina, at the point where d=0, or at 
the point where the normal to tlie neutral line is parallel to 
the direction of P x . 



356. Case of a rectangular beam. 

If the form of the beam be such that it may be divided 
into lamime parallel to ABCD of similar forms and equal 
dimensions, and if the pressure P x applied to each lamina 
may be conceived to be the same ; or if its section be a rec- 
tangle, and the pressures applied to it be applied (as they 
usually are) uniformly across its width, then will the distance 
k of the neutral line of each lamina from the centre of gra- 
vity of any cross section of that lamina, such as PT, be the 
same, in respect to corresponding points of all the laminae, 
whatever may be the deflection of the beam ; so that in this 
case the neutral surface is always a cylindrical surface. 



357. Case in which the deflecting pressure Y 1 is nearly per- 

pendicular to the length of the beam. 

In this case 4, and therefore sin. d, is exceeding small, so 
long as the deflexion is small at every point R of the neutral 
line ; so that h is exceedingly small, and the neutral line 
of the lamina passes very nearly, or accurately, through the 
centre of gravity of its section PT. 

358. The radius of curvature of the neutral surface 

OF A BEAM. 

Since the pressures applied to the portion APTD of the 



lamina ABCD are in equilibrium, the principle of the equality 
of moment.- must obtain in respect to them ; taking, there- 



RADIUS OF CURVATURE. 47l 

fore, the point R, where the neutral axis of the lamina inter- 
sects PT, as the point from which the moments are measured, 
and observing that the elastic pressures developed by the 
extension of the material in RP and its compression in RT 
both tend to turn the mass APTD in the same direction 
about the point R, and that each such pressure upon an 

element &k of the section PT is represented (equation 498) by 

■p 

:— -p4&, and therefore the moment of that pressure about the 

R 

■p 

point R by ^p 2 ^, it follows that the sum of the moments 
R 

about the point R of all these elastic pressures upon PT is 
represented by ^^p 2 ^, or by — , if I be taken to represent 

the moment of inertia of PT «bout R. Observing, moreover, 
that if p represent the length of the perpendicular let fall 
from R upon the direction of any pressure P applied to the 
portion APTD of the beam, Yjp will represent its moment, 
and sPjp will represent the sum of the moments of all the 
similar pressures applied to that portion of the beam ; 
we have by the principle of the equality of moments, 

359. The neutral surface of the beam is a cylindrical sur- 
face, whatever may be its deflection or the direction of its 
deflecting pressure, provided that its section is a rectangle 
(Art. 356.) ; or whatever may be its section, provided that its 
deflection be small, the direction of the deflecting pressure 
nearly perpendicular to its length, and its form before de- 
flexion symmetrical in respect to a plane perpendicular to the 
plane of deflexion. In every such case, therefore, the neutral 
lines of all the laminae similar to ABCD, into which the 
beam may be divided, will have equal radii of curvature at 
points similar to R lying in the same right line perpendicular 
to the plane of deflection; taking, therefore, equations simi- 
lar to the above in respect to all the laminae, multiplying 
both sides of each by I, adding them together, and observ- 
ing that R and E are the same in all, we have ~ r — ^ 
In this case, therefore, I may be taken in equation ^00) to 



17 "2 MOMENT OF INERTIA. 

represent the moment of inertia of the whole section of the 
beam, and P the pressure applied across its whole width. 



\e radius of curvature of a learn whose deflexion ts 
small, and the direction of the deflecting pressures nearly 



360. Th 
small, 
perpendicular to the length of the beam. 



In this case the neutral line is very nearly a straight line, 
perpendicular to the directions of the deflecting pressures ; 
so that, representing its length by o:. we have, in this case, 
j>=x ; and equation (500) becomes 

"5= El •••• (501 ) ; 

which relation obtains, whatever may be the form of the 
transverse section of the beam, I representing its moment of 
inertia in respect to an axis passing through its centre of 
gravity and perpendicular to the plane of deflexion. 



361. The moment of inertia I of the transverse section of a 
beam about thi centre of gravity of the section. 

In treating of the moments of inertia of bodies of different 
geometrical forms in a preceding part of this work (Art 82, 
&c. , we have considered them as solids; whereas the mo 
ment of inertia I of the section of a beam which enters into 
equation (500) and determines the curvature of the beam 
when deflected, is that of the geometrical area of the section. 
Knowing, however, the moment of inertia of a solid about 
any axis, whose Bection perpendicular to that axis is of a 
given geometrical form, we can evidently determine the 
moment of the area of that section about the same axis, by 
supposing the Bolid in the first place to become an exceed- 
ingly thin lamina i i. e. bv making that dimension of the 
solid which is parallel to the axis exceedingly small in the 
expression for the moment of inertia,), and then dividing 
the resulting expression by the exceedingly small thickness 
of this lamina. AVe shall thus obtain the" following values 



MOMENT OF INERTIA. 473 

362. For a beam with a rectangular section, ) 
whose breadth is represented by b and its depth VI= i l5c , 1 
by c (equation 61), ) 



363. For a beam with a triangular ) i c 
section, whose base is b and its height c rl = "oTj (i& 3 +i<? 3 )' 
(equation 63), ) 

364. For a beam or column with a circular ) y _ i * 
section, whose radius is c (equation Q6), f "~ * c * 



365. To determine the moment of inertia I in respect to a 
beam whose transverse section is of the 
form represented in the accompanying 
figure, about an axis ab passing through 
its centre of gravity ; let the breadth of 
the rectangle AB be represented by b 1 and 
its depth by d x , and let b 2 and d 2 be simi- 
n larly taken in respect to the rectangle EF, 
and b 3 and d 3 in respect to CD ; also let I x 
represent the moment of inertia of the section about the axis 
cd passing through the centre of CD, A l5 A 2 , A 3 , the areas 
of the rectangles respectively, and A the area of the whole 
section. 

Now the moments of inertia of the several rectangles, 
about axes passing through their centres of gravity, are 
represented by A&A*, i\b$*, T V Ws 8 > and the distances of 
these axes from the axis cd are respectively JO^i + ^a)? 
i{d 9 + d t ), 0. Therefore (equation 58), 

I x = T \bA* + i (<* i + ^s) 2 A, + fjbfi! + &d u + d 3 y A a + T vM 3 3 ; 

but A x =&/?!, A a =&X> A 3 =b 3 d 3 ; 

.-. I 1 = 1 V(AA a + A a £, 2 + A 3 d 3 >) + lid, + d 3 y A, +£(£ + d s y A a . 

Also if h represent the distance between the axes ab and cd* 
then (Art. 18) AA=J(^ 3 + ^ 3 )A 9 — K^ + ^A,, and (equation 
58) I^-A'A. 

/J= T V(A 1 ^ + A / 7 a 2 +A3<Z 3 3 )+^ 

t ^ "" {oVZ). 

If d l and d 2 be exceedingly small as compared with d tf 



c 


|B 


D 


— ■ d 
h 


l 


E 


r 



474 



1)1 I LKXION OF A BEAM. 



aeglecting their values in the two last terms of the equation 
and reducing, we obtain 

(503). 

If the areas AB and EF be equal in every respect, 
I^K + S^ + ^^ + AA,^ (504). 



366. The work expended upon the deflexion of a beam 
to which given pressures are applied. 

If aP represent the pressure which must have operated 

to produce the elongation or 
compression which the ele- 
mentary fibre pq receives, 
by reason of the deflexion 
of the beam, &x the length 
of that fibre before the de- 
flexion of the beam, and a& 
its section; then the work 
which must have been done 
upon it, thus to elongate 
or compress it, is repre- 
sented, equation (487) by 

Ep 
But (equation 498) aP=— r A&. The work ex- 




\ 



(A-py . A s? 
E.Aft * 



ponded upon the extension or compression of pq is there- 
in re represented by 

And the same being true of the work expended on the 
compression or extension of every other fibre composing the 
elementary solid VTPQ, it follows that the whole work 
expended upon the deflexion of that element of the beam 

Lb represented by -J^ r 2p a A&, or by i™Aa?; for 2p*A& repre- 
sents the moment of inertia I of the section PT, about an 
axis porpendieulartotheplaneof ABCD, and passing through 
the point R. If, therefore, a x be taken to represent the 
Length of that portion of the beam which lies between D 



DEFLEXION OF A BEAM. 475 

and M before its deflexion, and therefore the length of the 
portion ac of its neutral line after deflexion, then the whole 
woi'h expended upon the deflexion of the part AM of the 

beam is represented by i E2 ^&x. But (equation 500) ^= 

P v> a 

* p 1 ; whence, by substitution, the above expression 

becomes J-4r o^T Ax - Passing to the limit, and represent- 
ing the work expended upon the deflexion of the part AM 
of the beam by u t , 



u > = mJ x^ • • • • (506 >- 



367. The work expended upon the deflexion of a beam of 
uniform dimensions, when the deflecting pressures are 
nearly perpendicular to the surface of the beam. 

In this case I is constant, andj?j=:a?; whence we obtain 
by integrating (equation 505) be- 
tween the limits and a l5 

P 2 a 8 

^ = *~eT • • • * (506) ' 

where u x represents the work ex- 
pended upon the deflexion of the 
portion AM of the beam. Simi- 
larly, if bc=a ui the work expended 
upon the deflexion of the portion BM of the beam is repre- 
sented by 

P.V 

^i-^X ; 

so that the whole work U 3 expended upon the deflexion of 
the beam is represented by 




6EI 

But by the principle of the equality of moments, if a 

represent the whole length of the beam, 






A BEAM. 



P^=P =Pa. 

?, and ?. - .quations and the pre- 

btain by reduction 



r.=.— - - 



If the pressure P, be applied in the centre of the beam, 



- THE T.TVF»T> DEFl P A BEAM WHO THE DffiECnOJf 

OF THE DEF1ECTXSG PBESSEBE IS PEBPEXDICTT. A R TO ETS 



- tkm MN 



remain fixed, the deflexion taking 
place on either side of that section ; 
then ti l representing the work ex- 
pended upon the deflexion of the 
portion AM of tLe beam, and D l 
the deflexion of the point to which 
applied, measured in a direc- 
tion perpendicular to the surla; 

nation 40), u.= IF 
therefore P 1= T = -. 
But by equation (506 . — - = - -^r- \ Therefore P, = i^rrr ■ 
^ ; therefore ^ 




El 



whence we obtain by integration 



tf.'P, 



D «=W ' ' 



" 



If the whole work of deflecting the beam be done by the 

ire P,. the points of application of P and P, having: no 

mot: ..e direct: s 9E - Art 52 . . then 

eding in respect to equa* " before in 

to equation resenting the defir 



» Ci.:.'.." s I .5 .".. A:: 1". 



DEFLEXION OF A BEAM. 477 

perpendicular to the surface of the beam at the point of 
application of P 3 by D 3 , we shall obtain 

If the pressure P 8 be applied at the centre of the beam 

Eliminating P, between equations (506) and (509), and P, 
between equations (507) and (510), we obtain 

3EDV 3aEHV 

"*- 2< ' Us -2(^ 2 ) 2 ^' 

by which equations the work expended upon the deflexion 
of a beam is determined in terms of the deflexion itself, as 
by equations (506) and (507) it was determined in terms of 
the deflecting pressures. 



369. Conditions of the deflexion of a beam to which are 
applied three pressures, whose directions are nearly 
perpendicular to its surface. 

Let AB represent any lamina of the beam parallel to its 




plane of deflexion, and acb the neutral line of that lamina 
intersected by the direction of P 3 in the point c. 

Draw xx x parallel to the length of the beam before its 
deflexion, and take this line as the axis of the abscissa?, and 
the point c as the origin ; then, representing by x and y the 

* This result is identical with that obtained by a different method of inves* 
tigation by M. Navier {Resume de Lecons de Construction, Art. 859.). 



EQ1 

. and by E the radius of curva 
we have* 






ion of the beam being supposed exceed- 
inclination to ex of the tangent to the 

al line is. at all points, exceedingly small, so that l-^J 

may be neglected as compared with unity ; therefore pr=-^. 

- gtitating this value in equation (501), and observing that 
in this rented by (a 1 —x) instead 1 

the direction of the pressure P, being supposed nearly per 
..cular to the surface of the beam, and I constant. Let 
the above equation be integrated between the limits and 
_ ^en to represent the inclination of the tangent 

at c to ex. so that the value of -^ at c may be represented by 

tan. 

g-tan. .3=!^-^ . . . (514> 

Integrating a second time between the limits and a\ and 
observing that when 2=0, y=0, 

p 

y=^ l \ia 1 J-±a>\+xtaii. 5 . . . . (515). 

Proceeding similarly in respect to the portion he of the neu- 
tral line, but observing that in respect to this curve the value 

of — -j- at the point c is represented by tan. -3, we have 

d*'- EI ' 
g+tan. .3= j^-^'i . . (516). 

• Church's Diff. Cal. Art 



EQUATION TO THE NEUTRAL LINE. 479 

yJ^liarf-itfl-xtim. P . . . (517.) 

If D x and D 2 be taken to represent the deflexions at the 
points a and b, and ca and cb be assumed respectively equal 
to cd and ce, 

P a 3 
by equation (515), T> 1 = -^7-4-^ tan. (3, 

P a 3 
by equation (517), D s =-^y— # 2 tan. fi. 

If the pressures P x and P 2 be supplied by the resistances 
of fixed surfaces, then D X =D 2 . Subtracting the above equa- 
tion we obtain, on this supposition, 

0= F/ ^~ P ^ 3 +(^ + ^) tan. A 

Now T 1 a;-P 2 a:=^^^^=Y 3 a 1 a i (a-a i ); ob- 
serving that P I a=P 8 a 2 , P 2 «=P 3 « 1 , and a 1 -\-a i =a, 
. tan. |8= P - a ^-^) . . . (518). 

If /3 X , /3 2 represent the inclinations of the neutral line to 
xx x at the points a and &, then by equations (514) and (516) 

tan./3 -tan./3=^, tan. /3 3 + tan./3=^. 

Substituting for tan. /3 its value from equation (518), elimi- 
nating and reducing, 

tan.^_ 6EI ^ ,tan.p 3 - 6E ^ . ...<5±»). 

To determine the point m where the tangent to the neutral 
line is parallel to cxx n or to the undeflected position of the 

beam, we must assume >-=0 in equation (516)*; if we 

then substitute for tan. £ its value from equation (518), 
substitute for P 2 its value in terms of P 3 , and solve the 

* Church's Diff. Cal. Art. 78. 



4S0 



, I'll OF THE NEUTRAL LINE. 



resulting equation in respect to x, we shall obtain for the 
distance of the point m from c the expression 



a,+ Vi< a ,+ 2a d 



(520). 



370. The length of the neutral line, the beam being 
loaded ln the centre. 

Let the directions of the resistances upon the extremities 




of the beam be supposed nearly perpendicular to its surface ; 
then if x and y be the co-ordinates of the neutral line from 
the point a, we have (equation 501), representing the hori- 
zontal distance AB by 2a, and observing that in this case 

^p=— -33 j and that the resistance at A or B =JP, 

Integrating between the limits x and a, and observing that 
at the latter limit -4r =0, 

E1 !=i p ^-*')- 
Now if 5 represent the length of the curve ac, 

a a 



Church's Int. Cal. Art. 197. 



THE DEFLEXION OF A BEAM. 



481 






the deflexion being small, -j-j is exceedingly small at every 
point of the neutral line. 

.*. s — J | 1 -f- g2ET ^~ ^ a * x * + x *) \ d® > 



PV 

- 5 = ^+ 60ET * * ' ^ 521 > 



Eliminating P between this equation and equation (511), and 
representing the deflexion by D, 



8=a+i 



a ' 



371. A BEAM, ONE PORTION OF WHICH IS FIRMLY INSERTED IN 
MASONRY, AND WHICH SUSTAINS A LOAD UNIFORMLY DISTRI- 
BUTED OVER ITS REMAINING PORTION. 

Let the co-ordinates of the neutral line be measured from 



* The following experiments were made by Mr. Hatcher, superintendant of 
the work-shop at King's College, to verify this result, which is identical with 
that obtained by M. Navier (Resume des Lecons, Art. 86.). Wrought iron 
rollers "7 inch in diameter were placed loosely on wrought iron bars, the sur- 
faces of contact being smoothed with the file and well oiled. The bar to be 
tested had a square section, whose side was -7 inch, and was supported on the 
two rollers, which were adjusted to 10 feet apart (centre to centre) when the 
deflecting weight had been put on the bar. On removing the weights care- 
fully, the distance to which the rollers receded as the bar recovered its hori-. 
zontal position was noted. 



Deflecting Weight 
in lbs. 


Deflection in Inches. 


Distance through which 

each Roller receded 

in inches. 


Distance through which 

each Roller would have 

receded by Formula. 


56 
84 


3-7 
545 


•1 
•2 


•18 
•29 



31 



182 



THi: DEFLEXION OF A BEAM 




the point B where the beam 
is inserted in the masonry, 

and let the length of the 
portion AD which sustains 
the load be represented by 
(7, and the load upon each 
unit of its length by f* ; 
then, representing by x and 
y the co-ordinates of any 
point P of the neutral line, 
and observing that the pres- 
sures applied to AP, and in 
equilibrium, are the load v-(a-z) and the elastic forces 
developed npon the transverse section at P, we have by the 
principle of the equality of moments, taking P as the point 
from which the moments are measured, and observing that 
since the load f*(a — x) is uniformly distributed over AP it 
produces the same effect as though it were collected over the 
centre of that line, or at distance i{a—x) from P : observing, 
moreover, that the sum of the moments of the elastic forces 
upon the section at P, about that point, is represented 

358.) by -j>, or by EI ^ (Art. 369.) ; 

Elg=M«-*)'...(522). 

Integrating twice between the limits and x, and observing 
that when 85=0, 3r=0 and y=Q, since the portion BC of the 
beam is rigid, we obtain 

FA^=-i*(a-zy4-i*a>...(52S\ 

E3y=AKa— xy+}itfx—&w 4 - 

which is the equation to the nei\tral line. 

Let, now, a be substituted for x in the above equal 
and let it be observed that the corresponding value of y 
represents the deflexion D at the extremity A of the beam ; 
..all thus obtain by reduction 






M-fl 



D =8EI ' 



LOADED UNIFORMLY. 



483 



Representing by P tbe inclination to the horizon of the tan- 
gent to the neutral line at A, substituting a for x in equation 

dy 



(523), and observing that when x: 



dx 



= tan. /3, we obtain 



^a 



tan./3=^j (526). 



372. A BEAM SUPPORTED AT ITS EXTREMITIES AND SUSTAINING 
A LOAD UNIFORMLY DISTRIBUTED OYER ITS LENGTH. 




Let the length of the beam be represented by 2a, the load 
upon each unit of length by p ; take 
x and y as the co-ordinate of any 
point P of the neutral line, from the 
origin A; and let it be observed 
that the forces applied to AP, and in 
equilibrium, are the load \^x upon that 
portion of the beam, which may be 
supposed collected over its middle 
point, the resistance upon the point A, which is represented 
by M-#, and the elastic forces developed upon the section 
atP; then by Art. 360., 

EI^.=iw*-wx (527). 

Integrating this equation between the limits x and a, and 

observing that at the latter limit ^ = 0, since y evidently 

ax 

attains its maximum value at the middle C of the beam, 

. (528). 



EI ^ = i^x'-a^-iixaix'-a*) 



Integrating a second time between the limits and x, and 
observing that when a?3=0, y=0, 

Ely=^(W-a*x)-&a($x % -a*x) (529), 

which is the equation to the neutral line. Substituting a for 



184 Tin: DKFLEXIOH Of A BEAM 

x in this equation, and observing that the corresponding 
ralne of y represents the deflexion D in the centre of the 
beam, we have by reduction 

«-&■■■■** 

■•■senting by P the inclination to the horizon of the tan- 
gent to the neutral line at A or B, and observing that when 

x=0 in equation (528),^= tan. (3, 



*-'=S (531) - 

Let it be observed that the length of the beam, which in 
equation (511) is represented by a, is here represented by 
2v, and that equation (530) may be placed under the form 

u i (2a i 3 
D=f. — 77771 — ; whence it is apparent that the deflexion 

of a beam, when uniformly loaded throughout, is the same 
as though £ ths of that load (2a^) were suspended from its 
middle point. 



373. A BEAM IS SUPPORTED BY TWO STRUTS PLACED SYM- 
METRICALLY, AXD IT 18 LOADED UNIFORMLY THROUGHOUT 
ITS WHOLE LENGTH ; TO DETERMENT ITS DEFLEXION. 

Let CD=2a, CA=a i . load upon each foot of the length 
. of the beam=f* ; then load on 

each point of support = mi. Take 

^^^^^^^^^^^^^ C as the origin of the co-ordinates ; 

j* c then, observing that the forces 

impressed upon any portion OP 

I of the beam, terminating between 

C and A, are the elastic forces 
__^_ ■ — —ill- _, upon the transverse section of the 
beam at P, and the weight of the 
load upon CP; and observing that the weight f*CP of the 
load apon CP, produces the same effect as" though it were 
collected over the centre of that portion of the beam, so that 
iment about the point P is rej id by u.CP.-JCP, 



LOADED UNIFORMLY. 485 

or by ^CP"; we obtain for the equation to the neutral line 
in respect to the part CA of the beam (Art. 360) 
dry 
EI d5=i^ (532). 

Since, moreover, the forces impressed upon any portion CO 
of the beam, terminating between A and E, are the elastic 
forces developed upon the transverse section at Q the 
resistance w of the support at A, and the load_upon CQ, 
whose moment about Q is represented by J^CQ 1 , we have 
(equation 501), representing CQ by x, 

dS/ 
EI ^=4^— ^(^-^O ■ (533). 

Bepresenting the inclination to the horizon of the tangent to 
the neutral line at A by /3, dividing equation (532) by p. 
integrating it between the limits x and a» and observing 

that at the latter limit ^=tan. (3, we have, in respect to the 
portion CA of the beam, 






ar—±a. 



(534). 



Integrating equation (533) between the limits x and a, and 
observing that at the latter limit ^=0, since the neutral 
line at E is parallel to the horizon, 

Sidy 

Tfi* =iX -M3-ai)"-*a' + M«-aO a (535); 

which equation having reference to the portion AE of the 
beam, it is evident that when x=a„ ^=tan. (3. 

EI n 
^--tan.(3=ia(a-a i y-i(a 3 -a:)=}(a-aX2a' i ^4aa 1 -~.a^ 

.... (536). 

Substituting, therefore, for tan. (3 in equation (534), and 
reducing, that equation becomes 

Eldy , 

7 far* x +W«-a,)-K (537). 



Tin: DKFLEXIOJI OF A BEAM 

Integrating equation o the limits a, and x. and 

equation twees the limits and a?, and representing 

the deflexion at C, and therefore the value ofy at A, by D„ 

El 

~&f-D 1 ) = A^~*«(«~a.) , -IK-4«(*-«0 , J *-*«.*+ 

EI 

-^r=A^+ fM«-«jr-H^l« ( 538 ); 

the former of which equations determines the neutral line 
of the portion AE. and the latter that of the portion CA of 
the beam. Substituting a x for as in the latter, and observing 
that y then becomes D, ; then substituting this value of D x 
in the former equation, and reducing, 

Di=^{lM*-^"-(^--0} • • • • (539); 

EI 

— ]/=^ t -ia( : c-a i y + ia{S(a-a i y-a'\x (540). 

The latter equation being that to the neutral line of the por- 
tion AE of the beam, if we substitute a in it for ,/.', and 
represent the ordinate of the neutral line at E by y a we 
shall obtain by reduction 

y * = 2ffiT l^+^Xa-^-Sa'} .... (541). 

=0, or if the loading commence at the point A of the 
beam. thi f y x will be found to be that already deter- 

mined for the deflexion in this case (equation '•'■ 

N >w, representing the deflexion at E by D a , we have evi- 
dently T>.—D 1 —y l . 



374. Tup: conditions of the deflexion of a beam loaded 
uniformly throughout its length, and sitported at 
its extremities a and d, and at two points b and c 
situated at equal distances from them, and in the 
same horizontal straight line. 

Let AB=(/ : . AD=2a. 

Let A be taken as the origin of the co-ordinates; let the 



LOADED UNIFORMLY. 



487 




pressure upon that point be 
represented by P l5 and the 
pressure upon B by P 2 ; also 
the load upon each unit of 
the length of the beam by M-. 
If P be any point in the 
neutral line to the portion AB 
of the beam, whose co-ordi- 
nates are x and y, the pres- 
sures applied to AP, and in equilibrium, are the pressure 
P 2 at A, the load ^x supported by AP, and producing the 
same effect as though it were collected over- the centre of 
that portion of the beam, and the elastic forces developed 
upon the transverse section of the beam at P ; whence it 
follows (Art. 360.) by the principle of the equality of 
moments, taking P as the point from which the moments 
are measured, that 



EI-3=^ 



•p.* 



(543). 



Integrating this equation between the limits a iy and a?, and 
representing the inclination to the horizon of the tangent to 
the neutral line at B by /3 8 , 

El( j-tan. /3 a )=M^-<)-P , I (® , -«. s ) • • • • (544). 

Integrating again between the limits and x, 

E%-a>tan. P,)=}i>>(&-a l ty-%P l (ia?-a'x) . . . (545), 

Whence observing that when x=-a^ y=0, 

EI tan. P^twf-iPtf (546). 

Similarly observing, that if x and y be taken to represent 
the co-ordinates of a point Q in the beam between B and C, 
the pressures applied to AQ are the elastic forces upon the 
section at Q, the pressures P, and P 2 and the load i*>x } we 
have 

EI-^=i^-P 1 a?~P 9 (»-a J ) (547). 

Integrating this equation between the limits a x and x, and 
observing that at the former limit the value of tt is repre- 
sented by tan. /3 a , we have 



488 THE DEFLEXIOB OF A BEAM 

KI (J-tan. fi.) = JM <rf_a 1 •)_JP 1 (»'-a 1 ■)-iP,(a>-« 1 )■ 

.... (548). 

Now it is evident that, since the props B and C are placed 
symmetrically, the lowest point of the beam, and therefore 
of the neutral line, is in the middle, between B and C ; so 

that y = 0, when x=a. Making this substitution in equa- 
tion (548), 
-EI tan. /b'^X^-^-iPX^-O-iP,^-^) 5 • • (549). 

Since, moreover, the resistances at C and D are equal to 
those at B and A, and that the whole load upon the beam is 
sustained by these four resistances, we have 

P 1 + P 2 =^ (550). 

Assuming a^na, and eliminating P„ P 2 , tan. £„ between 
the equations (546), (549), and (550), we obtain 

_»* j *' + 12n'-24» + 8 \ 



, _W j 
3 ~8/i * ( 



4n*—n 9 -8 \ _ w{n-2) j — n' + Zn + l . 
272,-3 \~ $n I 2ti-3 »"^ 



a w'n ( 5n'— 21n* + 24/1-8 ) 

tan - ^m • I 2^3 [ = 

pa'njn— 1) ( 5?i 9 — 16ti + 8 ) (553^ 

24EI ' ( 2ti-3 J { } ' 

Making %=0 in equation (544); and observing that the cor- 
responding value of -T- is represented by tan. fi iy we have 

EI (tan. /3,-tan. ^)=-^ a i' + i P A i - 

Substituting for tan. £ a and P l their values from equations 
(553) and (551), and reducing, 

a w*n ( -3n 3 + 18^- 247i + 8 ( 
tan - ^=48EI ( 2^3 f * ' ' ' W 

Representing the greatest deflexions of the portions AB and 



LOADED UNIFORMLY. 



489 



BC of the beam, respectively, by D 1 and D 2 , and by a?, the 
distance from A at which the deflexion J) 1 is attained, we 
have, by equations (544) and (545), 

-EItan./3 2 =i f .(^-<)-iP i ( a?1 2 -^) | 

Elffl-x, tan. /3,)=Mi« 4 -«>i)-iI > i(*»x , -a> 1 ) J * ' { ] ' 

The value of D 1 is determined by eliminating x x between 
these equations, and substituting the values of P x and tan. /3 a 
from equations (551) and (553). 

Integrating equation (548) between the limits a y and a, 
and observing that at the latter limit y=D 2 , we have 

EID 2 =EI(a-<) tan. ^^^{^-a^-a^ia-a,)} - 

Substituting in this equation for the values of tan. /3 2 , P l5 P a , 
and reducing, we obtain 

D '= 4Am-L) K-2n'-8» + g{ (556). 

Representing BC by 2a 2 , and observing that a 9 = AE — 
AB = a — na = (1 — n)a, 



D n 



^ 2 4 n s — 2^ a — 8^+6 



48EI * (3-2n)(l-ny 



(557). 



375. A BEAM, HAVING A UNIFORM LOAD, SUPPORTED AT EACH 
EXTREMITY, AND BY A SINGLE STRUT IN THE MIDDLE. 

If, in the preceding article, a, be assumed equal to a, or 

71=1, the two props B and 
C will coincide in the centre ; 
and the pressure P a upon 
the single prop, resulting 
from their coincidence, will 
be represented by twice the 
corresponding value of P a in 
equation (552) ; we thus ob- 
tain 




P,=ff*a, Px=f^; 



pa, 



tan-ft=i8El> tan -^=°- 



(55S). 



490 THK DEFLEXI'N OF A BEAM 

Tlie distance a>, of the point i _ est deflexion of either 
portion of the beam from its extremities A or 1). and the 
amount D, of that greatest deflexion, are determined from 
equations (555). Making tan. 8^=0 in those equations, 
substituting for P, its value, solving the former in respect to 
ind the latter in respect to D„ we obtain 



x t = 1( /' '-' = 421 535a (559). 

Y) ^(g-gQ'^ + g) -25997f*a 4 

Dl== SsEI = 4SEI ^ 6 °)* 



376. A BEAM WHICH SUSTAINS A UNIFORM LOAD THROUGHOUT 
ITS WHOLE LENGTH, AND WHOSE EXTREMITIES ARE SO FIRMLY 
IMBEDDED LN A SOLED MASS OF MASONRY AS TO BECOME 
RIGID. 

Let the ratio of the lengths of the two portions AB and 
AE oi a beam, supported by two props (p. 487), be assumed 
to be such as will satisfy the condition bri 1 — 16fl +8=0 ; or, 
solving this equation, let 

7i=j(±± V'6) (561). 

The value of tan. 0, (equation 553) will then become 

zero ; so that when this re- 
^EE lation obtains, the neutral 
line will, at the point B, be 
parallel to the axis of the 
abscissae : or, in other words, 
the tangent to the neutral 
line at the point B will retain, 
after the deflexion of the beam, the position which it had 
before; i. e.. its position will be that which it would have 
lied if the beam had been, at that point, rigid. Now 
this condition of rigidity is precisely that which results from 
the insertion of the beam at its extremities in a mas 

mm . - shown in the accompanying figure; whence it 
follows that the deflexion in the middle of the beam is the 
same in the two cases. Taking, therefore, the negative sign 
in equation (561 . and substituting for n its value |(4— I 6) 
or "6202041 in equation (557), and observing that, in that 




SUPPORTED AT ANY NUMBER OF POINTS. 



491 



equation, 2<z 2 represents the distance BC in the accompany- 
ing figure, we obtain 



D B 



24EI 



(562). 



By a comparison of this equation with equation (530), it 
appears that the deflexion of a beam sustaining a pressure 
uniformly distributed over its whole length, and having its 
extremities prolonged and firmly imbedded, is only one-fifth 
of that which it would exhibit if its extremities were free* 

If the masonry which rests upon each inch of the portion 
AB of the beam be of the same weight as that which rests 
upon each inch of BC, the depth AB of the insertion of each 
end should equal "62 of AE, or about three ten: lis of the 
whole length of the beam. 



N 


c - 

I 

4 


e !.ia d| a I 1 

- Th 

* p - , — 1 

I 2 1— I- 



377. Conditions of the equilibrium of a beam supported at 
any number of points and deflected by given pressures. 

To simplify the investigation, let the points of support 
p | p | p ^ ABC be supposed to be three 

in number, and let the direc- 
tions of the pressures bisect 
the distances between them ; 
the same analysis which de- 
termines the conditions of the 
equilibrium in this case will 
be found applicable in the more general case. Let P„ P 3 , 
P 6 , be taken to represent the resistances of the several points 
of support, a l and « 2 the distances between them, P 2 P 4 the 
deflecting pressures, and x y the co-ordinates of any point in 
the neutral line from the origin B. Substituting in equation 

(500) for =p its value -^f , and observing that in respect to the 

J.w OlX 

portion BD of the beam 2l ) p=Y^a 1 —x)—'P 1 (a l —x), and 
that in respect to the portion DA of the beam, zFp= 
— 1^(0,-0?), we have for the differential equation to the 
neutral line between B and D 

* The following experiment was made by Mr. Hatcher to verify this result. 
A strip of deal fV in. byf^ in. was supported with its extremities resting 
loosely on rollers six feet apart, and was observed to deflect 1*2 inch in the 
middle by its own weight. The extremities were then made rigid by confining 
them between straightedges, and, the distance between the points of support 
remaining the Bame, the deflexion was observed to be "22 inch. The theory 

would have given it '2 I. 



BEAM £ :> AT ANY NUMBER OF POINTS. 



E£g=?,(ia,-»)--p,(a,-x) (563), 



between 1) and A 



El'^-P/^-z) . . . (564). 

Representing by S the inclination of the tangent at B to the 
he abscissae, and integrating the former of these 
equations twice between the limits 

i:i'^=:- : W\ ■•/,/■- ,-\,-l\( ai x-$a?) + EI tan. 3 .... (565); 

EIy=iP,(fc t^-ix'i-iYja^-lx^ + Eh; tan. 3 . . . (566). 

ituting Ja, for a in these equations, and representing 
by 1\ the value of y. and by 7 the inclination to the horizon 
01 the tangent at the point D, we obtain 

EI tan. -/^P^'-fP^ + EItan. 3 .... (567;, 
EED a =APA a — A^A'+P 3 ^ tan - P - • - - (563). 
Integrating equation (564) between the limits -^ and g 

EI^=-P 1 (a 1 a?-iaO+EI *"- r+fPA'- 

Eliminating tan. y between this equation and equation (567) 
and reducing, 

El'^= -P l( >,r-K)4-EI tan. P+iPA 1 . ■ ■ (569). 

g rating again between the limits ■— and a?, and elimi- 
nating the value of Dj from equation (568 . 
EIy=- J K'-'^Itan. P+fPAfr-APA? ■ (570) 

Now i r [a evident that the equation to the neutral line in 

respect to the portion CE of the beam, will be determined 

nring in the above equation P, and P 4 for P, and P, 

•tiv.-ly. 

Making this substitution in eqnatio . and writing 

-tan. 3 id the resulting equation ; then assum- 



BEAM SUPPORTED AT ANY NUMBER OF POINTS. 493 

ing x=a 1 in equation (570), and x=a 2 in the equation thus 
derived from it, and observing that y then becomes zero in 
both, we obtain 

0=-iPA 3 + 5 P^' + EIa, tan. /3, 

0=-iP 6 < + ¥ vP 4 <-EIa 2 tan. (3. 

Also, by the general conditions of the equilibrium of parallel 
pressures (Art. 15.), 

P^+PV^P^+JP,^, 

P 1 + P 3 + P 5 =P 2 + P 4 . 

Eliminating between these equations and the preceding, as- 
suming #! + «„— a, and reducing, we obtain 



_ -P,a 1 (8g, + 5g 1 )— 3P 4 a, ' 
1_ 16^ 



(571). 
(572). 



p -M p ^( 1+ S) +p '( l+ S)l---- (5T3 )- 

By equation (568), 

D '=76& i Pa(1«^+^)-»Pa' } (5T4)- 

Similarly, 

^TeSlsl Pa(16<*,+7«,)-9P A °} (575); 

tan - /3 = 4^eW) P (3«,«-)-3P^ [ = 1 |i a { P *~ P * t' 570 )- 
By equation (567), 

If a, be substituted for x in equation (569), and for P, and 
tan. (3 their values from equations (571) and (576); and if 
the inclination of the tangent at A to the axis of x be repre- 
sented by /Sy we shall obtain by reduction 



494 BEAM Bl PPOBTED AT ANY 2TUMBER OF POINTS. 

•""•■ 3 .=32ElJ P t a,'--pj ts Qa,+a,) \ (578). 

Similarly, if \. represent the inclination of the tangent at 
C to the axis of x. 



tajL ^=mH I P A'-PA(2«,+«,) [ (57 



'■')■ 



378. If the pressures P 2 and P 4 , and also the distances <z, 
4iid a 2 , be equal, 

P,=P.= , S ,P„ P s = V'I J „ tan. /3=0, tan. /3,=tan. 0,=-§g[. 

379. If the distances <z, and a 2 be equal, and P 4 =3P 2 , 
P.=iP« P,= VP„ P 6 =fP 2 , tan. /3^-j^p tan./3 , = » 

380. If a =a 2 and 3P 4 =13P 2 , Y,=0, P. ~ l P a , P 6 =fP 2 . 

* The following experiments were made by Mr. Hatcher to verify this result. 
The bar ACB, on which the experiment was to be tried, was supported on 
knife edges of wrought iron at A, C, and B, whose distances AC and CB were 
eich five feet. The angles of the knife edges were 90°, and the edges were 
o'Jed previous to the experiments. The weights were suspended at points D 




¥ 



"a 



aniE intermediate between the points of support. In measuring the angles 
of deflexion the instrument (which was a common weighted index-hand turn- 
ing on a centre in front of a graduated arc) was placed so that the angle c 
of lit' parallelogram of wood carrying the arc was just over the knife-edge B, 
the side ed of the parallelogram resting on the deflected bar. This position 
gave the angle at the point of support. 

Isl Experiment. — A bar of wrought iron half an inch square, being loaded 
at E with a weight of 18 lb. 13 oz., and at D with 52 lb. 3 oz., assumed a per- 
fectly horizontal position at B, as shown by the needle. The proportion of 
eights is '2-77 : 1. 

2d Experiment. — A bar -7 inches square, being loaded at E with a weight of 
87*8 Ib.,*nd at D with a weight of 112 1b., assumed a perfectly horizontal 
position at B. The weights were in this experiment accurately in the propor- 
tion :; : 1. 

Experiment. — A round bar, -75 inch in diameter, being loaded at E with 
87*8 Hi., and at D with 112 lb., showed a deviation from the horizontal position 
at li amounting to not more than 20'. The weights were in the proportion of 
8: 1. 

The influence of the weight of the bar is not taken into account. 



A BEAM DEFLECTED BY PRESSURES. 



495 



381. Curvature of a rectangular beam, the direction of 
the deflecting pressure and the amount of the de- 
flexion being any whatever. 

The moment of inertia I (Art. 358.) is to be taken, about 
an axis perpendicular to the plane of deflexion, and passing 
through the neutral line, the distance h of which neutral 
line from the centre of gravity of the section is determined 
by equation (499). 

Now T l jhc 3 representing (Art. 362.) the moment of inertia 
of the rectangular section of the beam about an axis pass- 
ing through its centre of gravity, it follows (Art. 79.) that 
the moment I about an axis parallel to this passing through 
a point at distance h from it is represented by 

Substituting, therefore, the value of h from equation 
(499), 

R 2 P* 



1= 



E'fo 



sin. 3 d+ T Vfo 8 (580). 



Substituting this value in equation (500), and reducing, 
1 UF.Mqp, 



R~"12R*P 1 'sin. s, d + EW ' 



(581). 



Draw ax parallel to the 
position of the beam be- 
fore deflexion ; take this 
line as the axis of the 
abscissae and a as the 
origin ; then j^ = Hm=Hn 
% +wm=MR cos. MRra + 

aM sin. Mam— y cos. Mam 
-fa? sin. Mam. 

Let, now, the inclination 
DaP x of the direction of 
P, to the normal at a be 
represented by d„ and the inclination Mat of the tangent to 

the neutral line at a to ax, by /3, ; then Mcm=^— &+£,)• 

.\p x =y sin. (^ + (3,) + % cos. ft + ft)- 
Substituting this value of p x in the preceding equation, 




496 A BEAU DEFLECTED r,v PBEB8X7BE8. 

1 1i>lM-7»-|.v»ii..(>\ + ^ l ) +gcoB.(< l +g l )} ,_ 

i; L2RT 1 , Bin. f a+E ^ w " 

where d represents (Art. 355.) the inclination ~Rqa of the 
normal at the point R to the direction of P,. 



3S2. Case in. which the d<fexion of the team is small. 

If the deflexion be small, and the inclination d„ of the 
direction of P, to the normal at its point of application, be 

greater than - ; then y sin. (0,4-/3,) is exceedingly small, 

and may be neglected as compared with x cos. (d, + /3,); in 
this case, moreover, 6 is, for all positions of R, very nearly 
equal to 6 V Neglecting, therefore, (3 1 as exceedingly small, 
we have 

1 lSPJEteECOS.*, 

R"~12R , P l 1 sin. ^, + EW ^ h 

Solving this equation, of two dimensions, in respect to p, and 
taking the greater root, 

1 6P, 

r=E£7{*cosJ,+ ^cos.^-^sin.^J (584). 



3S3. The woke expended upon the deflexion of a uni- 
form EECT ANGULAR BEAM, WHEN THE DEFLECTING PRES- 
BUBES ABE IN( XJNED AT ANY ANGLE GREATER THAN HALF A 
BIGHT ANGLE TO THE SURFACE OF THE BEAM. 

If u t represent work expended on the deflexion of the 
portion AM of the beam, then (equation 505) 



'~±eJ i ' 



2 T? 

out by equation (500) y=pi • p 



A. BEAM DEFLECTED .BY PRESSURES. 497 



.-.u^iF.f^dx (585). 



B 

llt K~~ Efo 3 \ x cos * 6 i + V x * cos « 'K— i<? sin. 2 ^} x cos. 6 1 ; 
by equation (584), observing that the deflexion being small, 
p x =x cos. b x very nearly. Now the value of =p (equation 
584) becomes impossible at the point where x cos. ^ becomes 
less than — —o sin. d ; the curvature of the neutral line com- 

mences therefore at that point, according to the hypotheses 
on which that equation is founded. Assuming, then, the 

corresponding value — -c tan. ^ of x to be represented by a>„ 

V 3 
the integral (equation 585) must be taken between the limits- 
#, and a xi instead of and a l ; 

m 

;, u x -= — hj-j— 2 1 {^ cos. O^x \/x* cos. \— id* sin. a ^} dx ;. 

/.«.=?^^4^-|^t a ii.'a i +(« 1 '-iaan.*a i ) l j'*(686)' 

And a similar expression being evidently obtained for the 
work expended in the deflexion of the portion BM of the 
beam, it follows, neglecting the term involving c 3 as exceed- 
ingly small when compared with a^ that the whole work JJ l 
expended upon the deflexion is represented by the equation 

U,= ES? { Pl " cos - ^ S^+K-^ 3 tan - M 0** + 

P 2 a cos. % \a: + (a;-ic> tan. ^) l ) j . 

But if d 3 be taken to represent the inclination of P 3 to the 
normal to the surface of the beam, as d, and 0, represent the 
similar inclinations of F 1 and P 2 , tlien, the deflexion being 
small, 

* Church's Int. Cal. Art. 149. 

32 



498 A BEAM DEFLECTED BY PRESSURES. 



P 3 rt cos. ^=P 3 <2 3 cos. 6„ P 2 # cos. & 7 =Y,a 1 cos. 6 V 

Eliminating P, and P a between these equations and the 
preceding, 

g. ^jg£ft { O^' + K "^tan. *,)*} + 
a;\a: + {a;-lc\M.\y\ I (587). 

If the pressure P s be applied perpendicularly in the centre 
of the beam, and the pressures P, and P, be applied at its 
extremities in directions equally inclined to its surface ; then 
a l =a 9 =ia, 6^=6^=6, and d,=0. Substituting these values 
in the preceding equations, and reducing, 






16Ebc 



(588). 



384. The linear deflexion of a rectangular beam. 

D, being taken as before (Art. 368.) to represent the de- 
flexion of the extremity A measured in a direction perpen- 
dicular to the surface of the beam, we have (Art. 52.) 



u x — f? x cos. ^dJ), 



/.P, COS. 6 X : 



du, du x dP, 



But by equation (586), neglecting the term involving c% 

jp op 1 

.•.P,co S .4,=g . m tcos.'\\a l > + {a 1 '-tc<t l m.'6 i y\ 

Dividing both sides by P„ reducing, and integrating, 

op 3 

D^^cos-^Ja^ + ^-^tan. ^) 2 J .... (589) 

Proceeding similarly in respect to the deflection D' perpen 



INCLINED AT ANY ANGLE TO ITS SURFACE. 



499 



dicnlar to the surface of the beam at the point of application 
ol r s , we obtain from equation (587) 

-n 2P 3 cos. 0. ( , , 3 

3=: ~Ea7bc^~' a * K +(a l , -Jc"tan. \f\ + 



a*;&*+(*?~-W tanl %)h 



(590) 



In the case in which P, and P 3 are equally inclined to the 
extremities of the beam and the direction of P bisects it 
this equation becomes 3 ' 



_ P : > 3 +(a 2 -^ 2 tan. W\ 
3 $Eb? : 



(591). 



385. The work expended upon the deflexion of a beam sub- 
jected to the action of pressures applied to its extremities 
and to a single intervening point, and also to the action 
of a system of parallel pressures uniformly distributed 
over its length. 

Let «, represent the aggregate amount of the parallel 




pressures distributed over each unit of the length of the 
beam, and a their common inclination to the perpendicular 
to the surface ; then will p® represent the aggregate of those 
distributed uniformly over the surface D% and these will 
manifestly produce the same effect as though they were 
collected in the centre of DT. Their moment about the 
point K is therefore represented by w\x cos. a, or by tw? 
cos a ; and the sum of the moments of the pressures applied 
to AT is represented by (P,x cos. 0,-*^ cos. a). Substi- 
tuting this value of the sum of the moments for P v in 
equation (505), we obtain 

J-±l ?( F > X C0S - & -JW* COS. a)' 

U >~2EJ ' — ^ 



500 



DEFLEXION OF A BEAM BY PRESSURES. 



3SG. It tl<> pn 88ures he all perpendicular to the surface of 
tl,. beam, *,=0, a = 0, and I is constant (equation 499); 
whence we obtain, by integration and reduction, 



0, 



*=^»P. g -iP/*+*rtfl 



(592). 



It* the pressure P s be applied in the centre of the beam, 
P^iPs + J^a, and a 1 =^a i also the whole work U, of 
deflecting the beam is equal to 2u t ; whence, substituting 
and reducing, 



U .=4^» P . 1 + * I >* + *' aV } 



(593). 



387. A RECTANGULAR BEAM IS SUPPORTED AT ITS EXTREMITIES 
BY TWO FIXED SURFACES, AND LOADED LN THE MIDDLE I IT 
IS REQUIRED TO DETERMINE THE DEFLEXION, THE FRICTION 
OF THE SURFACES ON WHICH THE EXTREMITIES REST BEING 
TAKEN INTO ACCOUNT. 

It is evident that the work which produces the deflexion 




of the beam is done upon it partly by the deflecting pressure 
P, and partly by the friction of the surface of the beam 
upon the fixed points A and B, over which it moves whilst 
in the act of deflecting. Representing by z the limiting 
angle of resistance between the surface of the beam and 
either of the surfaces upon which its extremity rests, the 
friction Q t or Q 2 upon either extremity will be represented 
by £P tan. <p ; and representing by s the length of the 
curve oa or c&, and by '2 a the horizontal distance between 
the points of support; the space through which the surface 
of the beam would have moved over each of its points of 
support, if the point of support had been in the neutral line, 
is represented by s— a, and therefore the whole work done 
no. -n the beam by the friction of each point of support by 

i tan. fPd». Moreover, D representing the deflexion of 



THE SOLID OF THE STRONGEST FORM. 501 

the beam under any pressure P, the whole work done by P 

is represented by JYdT). Substituting, therefore, for the 

work expended upon the elastic forces opposed to the 
deflexion of the beam its value from equation (588), and ob- 
serving that the directions of the resistances at A and B are 
inclined to the normals at those points at angles equal to 
the limiting angle of resistance, we have 

J P^D + tan. ,j Yds= \ 6E ^ "■ 

But f VdD = f?^dP; 8indfFds=fF~d¥= 

oTypara / P 3 <#P by equation (521). 

Substituting these values in the above equation, and dif- 
ferentiating in respect to P, we have 

v dD_ P |a 3 + (a 2 -fc 2 tan. 2 cp)i} PV 

Dividing by P, and integrating in respect to P, 
Pja' + ^-jc- tan. ■«>)*} PV 



388. The solid of the strongest form with a given 
quantity of material. 

The strongest form which can be given to a solid body in 
the formation of which a given quantity of material is to be 
used, and to which the strain is to be applied under given 
circumstances, is that form which renders it equally liable to 
rupture at every point. So that when, by increasing the 
strain to its utmost limit, the solid is brought into the state 
bordering upon rupture at one point, it may be in the state 
bordering upon rupture at every other point. For let it be 
supposed to be constructed of any other form,, so that its 
rupture may be about to take place at one point when it is 
not about to take place at another point, then may a portion 
of the material evidently be removed from thesecond point 
without placing the solid there in the Btate bordering upon 



THE BCPTTRE OF A BAB. 

nipt added at the first poi: ike it out 

of the state bordering upon ruptu and thus 

I being no longer in t. bordering upon 

rupt ; be made to bear a strain g: 

than that which was before upon the point of breaking it, 
and will have been rendered stronger than it 
The first form was not therefore thee _ form of which 
it could have i:h the given quant: 

ial ; nor is any form the strongest which doo- 
the condition of an ' :> rupture at every 

The strongest form, with a given 
quar. . given m _ - .ngth 
under a g :ch can be con- 
strue- _ :h the least materia 
that the strongest form is also the form of the greatest 
- 

EUPTTBE. 

>. The rupture of a bar of wood or metal may take 
place either by a strain or tension in the direction of ita 
length, to which is o: r by a thrust or 

compressing force in tlie direction - I -ngth*, to which is 

oppos sistj bo C MPSSSBKHr; or each of these 

stance may oppose them- Ives 1 its rupture 
transversely, the one being called into operation on on 

. and "the other on the other side, as in the ca- 
a Tbassvebse S 



D wa :tt. 

of different materials as they have 
been determi: best authorities, and by the mean 

results of nu: 3 rrmente, will be found stated in a 

table at the end of this volume. The unit of tenacity is that 
opposed to the tearing asunder of a 1 a .uare inch in 

n, and is estimated in pounds. 1 ident that the 

tena« such bars placed side I 

a d section, would be equal 
such mntB, .... - :he tenacity of one bar. 

lind, th* the tenacity of a bar material 

in pounds, multiply the numhv. : - » sec- 



RUPTURE OF A BAR SUSPENDED VERTICALLY. 50S 

tion by its tenacity per square inch, as shown by the 
table. 



391. A BAR, CORD, OR CHAIN IS SUSPENDED VERTICALLY, CAR- 
RYING- A WEIGHT AT ITS EXTREMITY : TO DETERMINE THE 
CONDITIONS OF ITS RUPTURE. 

First. Let the bar be conceived to have a uniform section 
represented in square inches by K ; let its length in inches 
be L, the weight of each cubic inch fx, the weight suspended 
from its extremity "W, the tenacity of its material per square 
inch r ; and let it be supposed capable of bearing m times 
the strain to which it is subjected. The weight of the bar 
will then be represented by f^LK, and the strain upon its 
highest section by ^LK-f W. Now the strain on this section 
is evidently greater than that on any other ; it is therefore at 
this section that the rupture will take place. But the resist- 
ance opposed to its rupture is represented by Kr ; whence it 
follows (since this resistance is m times the strain) that 

Kr=m(fxLK+W), 

•■*=~L m. 

By which equation is determined the uniform section K of a 
bar, cord, or chain, so that being of a given length it may be 
capable of bearing a strain m times greater than that to 
which it is actually subjected when suspended vertically. 

The weight W x of the bar is represented by the formula 
KLfs 

•••».=5» m- 

392. Secondly. Let the section of the rod be variable ; and 
let this variation of the section be such that its strength, at 
every point, may be that which would cause it to beat-, 
without breaking, m times as great a strain as that which it 
actually bears there. Let K represent this section at a point 
whose distance from the extremity which carries the weight 
W is x ; then will the weight of the rod beneath that point 

be represented by I ^Kdx ; or, supposing the specific gravity 



504 ETTPTUEE OF A BAR SUSPENDED VEETICALI.Y. 

of the material to be every where the same, by ^f^Kdx : also 
the resistance of this section to rupture is Kr. 

/. m(W+ppKdx)='KT 

Differentiating this expression in respect to x, observing that 
K is a function of x, and dividing by B>, we obtain 

1 dK _ m^ m 
E dx ~~ T' y 

Integrating this expression between the limits and a?, and 
representing by Ko the area of the lowest section of the rod, 

But the strain sustained by the section Ko is W, therefore 
K T=m¥ ; 

:. K= — e r (597). 

The whole weight W 2 of the rod, cord, or chain, is repre- 
sented by the formula 

W 2 =ifKdx=^-feT X dx=W\e—-l) - - • (598). 

A rope or chain, constructed according to these conditions, 
is evidently as strong as the rope or chain of uniform section 
whose weight W, is determined by equation (596), the value 
of m being taken the same in both cases. The saving of ma- 
terial effected by giving to the cord or chain a section vary- 
ing according to the law determined by equation (598) is 
represented by 'W 1 — W 2 , or by the formula 

^-M^A- w> 

* Church's Int. Cal. Art. 159. 



THE SUSPENSION BRIDGE. 



505 



The suspension bridge. 

393. General conditions of the equilibrium of a loaded 

chain. 




Let AEH represent a chain or cord hanging freely from 

two fixed points A and H, 
and having certain weights 
w x , w 2 , 10,, &c, suspended by 
rods or cords from given 
points B, C, D, &c, in its 
length. Through the lowest 
point E of the chain draw 
the vertical Ea, containing 
as many equal parts as there 
are units in the weight of 
the chain between E and any 
point of suspension B, to- 
gether with the suspending 
rods attached to it, and the weights which they severally 
carry ; draw aP parallel to the direction of a tangent to the 
curve at B, and produce the tangent at E to meet aP in P ; 
then will aP and EP contain as many equal parts as there 
are units in the tensions at B and E respectively ; and if E£ 
and Ec be taken to represent the whole weights sustained by 
EC and ED, and Pb and Pc be joined, these lines will in 
like manner represent the tensions upon the points C and D. 
For the pressures applied to EB, and in equilibrium, being 
the weight of the chain, the weights of the suspending rods, 
the weights attached to the rods, and the tensions upon B 
and E, the principle of the polygon of pressures (Art. 9.) 
obtains in respect to these pressures. Now the lines drawn 
to complete this polygon, parallel to the weights, form 
together the vertical line Ea, and the polygon (resolving 
itself into a triangle) is completed by the lines aP and EP 
drawn parallel to the tensions upon B and E. Each line 
contains, therefore, as many equal parts (A»l. 9.) as there 
are units in the corresponding tension. Also, the pressures 
applied to the portion EC of the curve, being the weights 
whose aggregate is represented by FA and the tensions upon 
E and 0, of which the former is represented in direction 
and amount by EP, it follows (Art 9.) that the hitler is 

represented also in direction and amoum by the line IA 



506 



TIIK CATKNABY. 



which completes the triangle a¥b\ so that bV is parallel tc 
the tangent at C. 

In like manner it is evident that the tension upon D is 
represented In magnitude and direction by cP; so that cP is 
parallel to the tangent to the curve at D. 



The catenaky. 

394. If a chain of vniform section he suspended freely 
between two fixed points A and B, being acted upon by no 
other pressures than the weights of its parts, then it will 
assiuae the geometrical form of a curve called the 
oatenoyry. 

Let PT be a tangent to any point P of the curve inter- 
secting the vertical CD passing through its lowest point D 




in T ; draw the horizontal line DM intersecting PT in Q ; 
take this line as the axis of the abscissae ; and let DM =#, 
MP=y, DP=s, weight of each unit in the length of the 
chain =f*, tension at D=<?. Now DT being taken to repre- 
sent the weight ps of DP, it has been shown (Art. 393.) 
that DQ will represent the tension c at D, and TQ that 
at P. 

Also, ^= tan. PQM = tan. DQT =»^=y , 



dy 
dx 

dtf\ 



JXS 

' c 



. dx i dy\-* I 



14-^i 



. . (600). 

—j- J . integrating be- 



THE CATENARY. 507 

tween the limits and $,* and observing that when s=0, 

-Mi* 



W I. t ffrV.V* — A«c 



By addition and reduction, 



■s 






Substituting this value for s in equation (600), and inte- 
grating between the limits and x, 



v~- 




/ WD —\^x 

G G 


> 


■4 


f W 

2g 

IS — s 


2c 


hi 


2h is 


the equation 


to th 


3 catenary. 





) 



(603); 



395. The tension (c) on the lowest point of the catenary. 

Let 2S represent the whole length of the chain, and 2a 
the horizontal distance between the points of attachment. 
Now when x=a, s=S ; therefore (equation 602), 



S=£| T T (604); 



7_~J 



for which expression the value of c may be determined by 
approximation. 



396. The tension at any point of the chain. 
The tension T at P is represented by TQ= ^DQ' + ISP; 

* Church's Int. Cal. Art. 144. 



508 THE CATENARY. 

Now the value of c has been determined in the preceding 
article ; the tension upon any point of the chain whose dis- 
tance from its lowest point is s is therefore known. 



307. The inclination of the curve to the vertical at any 

point. 

CL1J 

Let i represent this inclination, then cot. l=-t ; 



/_, • ) 



/.(equation 600) cot. i=£l c c J (606). 



The inclination may be determined without having first 

un- 
determined the value of c, by substituting cot. i for — in 

equation (601) ; we thus obtain, writing also a and S for x 
and s, 



o=tan. i log. (cot. i + cosec. t)=tan. t log. cot. \i\ 

/. —tan. i log. tan. Ji=« (607). 

Tliis equation may readily be solved by approximation ; and 
the value of c may then be determined by the equation 
c=,aS tan. i. 



398. JL cAawVi of given length being suspended between two 
given points in the same horizontal line: to determine the 
depth of the lowest point beneath the points of attachment $ 
and, conversely, to determine the length of the chain whose 
lowest point shall hang at a given depth below its points 
of attachment. 



The same notation being taken as before 

dx 
dy 



t=(n-fr=(^r=e+'-'» 



THE CATENARY. 509 

Integrating between the limits and s, and observing that 
y=0 when s=0, 

y=l{(c'+^y-c] (608). 

Solving this equation in respect to *, 



Vy(y + j) 



s=V y\y + ~) (609). 

If H represent the depth of the lowest point, or the versed 
sine of the curve, then y=H when *=S. 

R=h(o'+^S')i-o} (610). 

r 1 



A(H+f) 



S=V H H+- (611). 



399. The centre of gravity of the catenary. 

If G represent the height of the centre of gravity above 
the lowest point, we have (Art. 32.) 



S.G=fyds=fy£d X . 

ds 

Substituting, therefore, for y and — - their values from equa- 



tions (602) and (603), we have 

a 

a 
c f* ( 2/xx —2/ix i fix -fix, \ 

HT ( s +s +2-2 ^ +s M 

C i C I 2^a -2//a\ 2c I J^ -f*a \ ) 

= Wl c . • ',„„ "TV . c )\ 



[ e — s -f2a s — e 



510 THE SUSPENSION BRIDGE 



s 4-s —4' 



But by equation (604) S=q"( c ~c~\ an( ^ D . v equation 
603), 

,. G=iJH- = (l--|)f (613). 



400. The suspension bridge of greatest strength, the 
weight of the suspending rods being neglected. 

Let ADB represent the chain, EF the road-way ; and let 



the weight of a bar of the material of the chain, one square 
inch in section and one foot long, be represented by m»„ the 
weight of each foot in the length of the road-way by fx 2 , the 
aggregate section of the chains at any point P (in square 
inches) by K, the co-ordinates DM and MP of P by x and y, 
and the length of the portion DP of the chain by 8. Then 

will the weight of DP be represented by f* x I Kds, and the 

weight of the portion CM of the roadway by ^„x\ so that 
the whole load (u) borne by the portion DP of the chain 
will be represented (neglecting the weight of the suspending 
rods) by 

M-, / Kds + pJC, .*. u=p / Kds + pjc (614). 



OF GREATEST STRENGTH. 511 

Let this load (V), supported by the portion DP of the 
chain, be represented by the line Da, and draw Dp in the 
direction of a tangent at D, representing on the same scale 
the tension c at that point ; then will ap be parallel to a 
tangent to the chain at P (Art. 393). 

•••!=; <■* 

ISTow let it be assumed that the aggregate section of the 
chains is made so to vary its dimensions, that their strength 
may at every point be equal to m times the strain which 
they have there to sustain. But this strain is represented in 
magnitude by the line ap (Art. 393.), or by (^-h^ 2 )*; if, 
therefore, r be taken to represent the tenacity of the mate- 
rial of the chain, per square inch of the section, then 

Er=m{c*+u*)* (616). 

(u*\* I dy* v* 

1 H — a) = mc ll + tt (equation 615) 

=mc-j- : therefore -7-= — . Also / Kds= I K~rdx= 
ax 7 ax mo J J dx 

— jK*dx=™J(c* + u i )dx (equation 616) ; 

/.(equation 614)w=— ^ / {c* + u*)dx+i*>jB. 

Differentiating in respect to x, and observing that -j- = 

du dy u du . . _ ^ . 

•7- -f= — 7- (equation 615), we have 
ay dx c dy ^ J 



u du m^ m^ / rc^\ 

c dy tg x } a re \ mi\ 



du u du rapt-j 
dx 



du r C udu 



re C du t C 
\x— / , y— / 






512 THE BU6PKN8ION BRIDGE 

Integrating these expressions,* we obtain 

X= ^U + Ifl^tan." 1 ^ + !»)-*« (617). 



u* + c* 






m'j. 



log. ^ -' [. 



'SmPi °' s a rc,a 



Substituting in this equation the value of u given by the 
preceding equation, and reducing, 



y 



^KHl^ + ^-l---^ 61 ^ 



which is the equation to the susjiension chain of uniform 
strength, and therefore of the greatest strength with a 

GIVEN QUANTITY OF MATERIAL. 



401. To determine the variation of the section K of the 
chain of the suspension bridge of the greatest strength. 

Let the value of u determined by equation (617) be sub- 
stituted in equation (616) ; we shall thus obtain by reduction 

K=™ \l+(l + ^)tan.^(l + ^-fx \* . (619). + 
r I \ mqt-J r \ cmpj ) J 

It is evident from this expression that the area of the sec- 
tion of the chains, of the suspension bridge of uniform 
strength, and therefore of the greatest economy of material, 
increases from the lowest point towards the points of suspen 
si on, where it is greatest. 



* Church's Int. Cal. Art. 133, Case IV. 

+ ~r= — : .•.«=— /Kdx. Now the function K (equation 619) mar be 
1 ax mc mcj 

integrated in respect to x by known rules of the integral calculus, the value 
of 8 may therefore be determined in terms of x, and thence the length in 
tonus of the span. The formula is omitted by reason of its length. 
Church's Int. Cal. Art. 129, Case II. 



OF GREATEST STRENGTH. 



513 



402. To determine the weight W of the chain of the suspen- 
sion oridge of the greatest strength. 

Let it be observed that W=:^ I ~Kds=u—^ i x (equation 

614) ; substituting the value of u from equation (617), we 
have 

T -( 1 +3tM ! ?K3i)''}-M"-<»' 



403. To determine the tension c upon the lowest point D of 
the chain of uniform strength. 

Let H be taken to represent the depth of the lowest point 
D, beneath the points of suspension, and 2a the horizontal 
distance of those points : and let it be observed that H and 
a are corresponding values of y and x (equation 618) ; 



.•.Ji= log. sec. \ — 1 -\l-\ a 

m^ e { r \ cmpj 

Solving this equation in respect to <?, 

e= I!±\ p_ S ec.-Vr H ) -lp. • • • (621).. 



404. The suspension bridge of greatest strength, the 
weight of the suspending rods being taken into ac- 
COUNT. 

Conceive the suspending rods to be replaced by a con- 




5H 



Till: SUSPENSION BRIDGE 



tinuous flexible lamina or plate connecting the roadway with 
the chain, and of snch a uniform thickness that the material 
contained in it may be precisely equal in weight to the ma- 
terial of the suspending rods. It is evident that the condi- 
tions of the equilibrium will, on this hypothesis, be very 
nearly the same as in the actual case. Let ^ s represent the 

weight of each square foot of this plate, then will m- 3 / ydx 

represent the weight of that portion of it which is suspended 
from the portion DP of the chain, and the whole load wupon 
that portion of the chain will be represented by 

u=pjKds-j-^x + ^/ydx .... (622). 

It may be shown, as before (Art. 400.), that 



dy_ u ir 



dx 



Kr=m{c' + uy 



(623). 



/ Kds=— I (c*-\- u*)dx. Substituting in equation (622), 
•differentiating in respect to x 9 and observing that -7-=- -7-, 



du u du mf*., 2 , 2X , 
Transposing, reducing, and assuming, 



(624). 



m^, 



(625); 



dy 



— 2a^ a = 2c(fA 3 y + a.C + f* 2 ). 



A linear equation in v?, the integration of which by a well 
known method gives 

—2 ay p -2 ay 

ire =z2cf{^ 3 y + 0LC + ^)e dy + C* 

Assuming the length of the shortest connecting rod DC to 
"he represented by //.integrating between the limits b and y. 
and observing that when y=b, u—0, 

* Church's Int. Cal. Art. 176. 



OF GREATEST STRENGTH. 515 

— lay c i I —lab —2ay\ ,i^ i / -2a7> —2ay^ j* 

U * E a^ 3 \ ~ ye ) 4 ' 2^ + aC + fX2 ) \ £ _£ ) f ' 

: ^JL J ^ _ y)+ (^+«i+is)(. -1) [ .(626). 

Substituting this value of u* in equation (623), and 
reducing, 

K =ir i te+^ +ao+ 4 -<*#-£-*■ ( -( 627 )5 

by which expression the variation of the section of the chain 
of uniform strength is determined. 

Differentiating the equation -^=- in respect to x, and 
substituting for j- its value from equation (624). 

Substituting for w' its value from equation (626), 
f ?« (It, ■ \ *«»-» (», 

Multiplying both sides of this equation by -^r, and integrat- 
ing between the limits h and y, observing that when y=b, 

'dy\ 2 /u \ 2c(y-&) 



ac 



W=(e+^+- + ^)<' -i)-^-»)- 



Now let it be observed, that the value of r, being in all 
practical cases exceedingly great as compared with the 
values of ^ and m, the value of a (equation 625) is exceed- 
ingly small ; so that we may, without sensible error, assume 
those terms of the series e 2a (y-&) which involve powers of 
2a(y— b) above the first, to vanish as compared with unity 

* Church'e Int. Cal. Art. 140. 



516 



THE SUSPENSION BRIDGE. 



This supposition being made, we have s 2a (y~ b )— l=2a(y— b), 
whence, by substitution and reduction, 



m 



:2{^h + CtC + ^) {y-b). 



Extracting the square root of both sides, transposing, and 
integrating, 



x 



the equation to a parabola whose vertex is in D, and its 
axis vertical.* 

The values a and H of x and y at the points of suspension 
being substituted in this equation, and it being solved in 
i espect to c, we obtain 



f* a +M 



2R-2b-aa 



i)a 3 



(629); 



by which expression the tension c upon the lowest point of 
the curve is determined, and thence the length y of the sus- 
pending rod at any given distance x from the centre of the 
span, by equation (628), and the section K of the chain at 
that point by equation (627), which last equation gives by a 
reduction similar to the above 



^1%-^H-W 1 



(630). 



405. The section of the chains being of icniform dimensions, 
as in the common suspension bridge, it is required- to 
determine the conditions of the equilibrium.^ 

The weight of the suspending rods being neglected, and 
the same notation being adopted as in the preceding arti- 
cles, except that ^ is taken to represent the weight of one 
foot in the length of the chains instead of a bar one square 
inch in section, we have by equation (614), since K is here 
constant, 

u=^s + ^x (631). 

* Church's Analyt. Geom. Art. 191. 

f This problem appears first to have been investigated by Mr. Hodgkinson 
in the fifth volume of the Manchester Transactions ; his investigation extends 
to the case in which the influence of the weights of the suspending rods is 
included. 



THE COMMON SUSPENSION BRIDGE. 01 1 

Differentiating this equation in respect to x, and observing 
that -T-=(l 4- -p) =(l + ^ij (equation 615), and that 

du du dy __du u _ i iA* 
dx=dy daT~dy ~e "^ \ + ?/ +f " a ; 

/^^ r udu 

Tlie former of these equations may be rationalised by 
assuming (c 2 + ^ 2 ) f = c + sw, and the latter by assuming 
(c* + w 9 )*:=3 ; there will thus be obtained by reduction 

/ (l+g>fe _ /» safe 

*- 2 y (l-^i^+^+o*-^!^"/ ^+w 

The latter equation may be placed under the form 

y=l/jl &_}*; 

which expression being integrated and its value substituted 
for 2, we obtain 

y= *l j („'+«.)»_,- ^ log. ^+y+^ I . . . (632) . 

Tlie method of rational fractions (Church's Integ. Calc. 
Art. 135) being applied to the function under the integral 
sign in the former equation, it becomes 






dz. 



Tlie integral in the first term in this expression Is repre- 
sented by $ log., |j j , and that of the second term by 



r— a — =-=r-: tan. 



or 



518 



KIPUKI. BY 



_js__ 1 (f*.+*s)»+(is— f»,)»g 

is g ater or Leee than i*,, or according as the 
weight of each foot in the length of the chains is greater <»r 
than the weight of each foot in the length of the road- 
Substituting for a its value, we obtain, therefore, in the 
two a - 8, 



-i 



log. 



2« 2 



(»+c)- •- - . (#*."—/*.')» 



tau 



g)'l('^H( 



M 



-H" 5 -k'H 



(633). 






If the given values, a and 1'. md y at the pointe 

suspension, be substituted in equations 
equations will be obtained, whence the value of the cone 
c and of u at the points of suspension may he determined by 
approximation. A series of values of w, diminishing from 
the value thus found to zero, being substituted in equations 
2 . as many corresponding values of x and y 
will then become known. -The curve 01 the chains may thus 
be laid down with any required degree of accuracy. 

This common method of construction, which assigns a 
uniform Bection to the chains, is evidently false in principle; 
the strength of a bridge, the section of whose chains varied 
according to the law established in Art. 401. (equation 619), 
would be far greater, the same quantity of iron I 
employed in its construction. 



RUTTUBE BY COMPRESSION, « 

'■. It results from the experiments of Mr. Eaton II 
kinson,* on the compression of short columns of different 
heights but of equal sections, first, that after a certain height 
is passed the crushing pressure remains the same, as the 



* Seventh Report of the British Association of Science. 



RUPTURE BY COMPRESSION. 519 

heights are increased, until another height is attained, when 
they begin to break ; not as they have done before, by the 
sliding of one portion upon a subjacent portion, but by 
bending. Secondly, that the plane of rupture is always 
inclined at the same constant angle to the base of the 
column, when its height is between these limits. These two 
facts explain one another ; for if K represent the transverse 
section of the column in square inches, and a the constant 
inclination of the plane of rupture to the base, then will 
K sec. a represent the area of the plane of rupture. So that 
if 7 represent the resistance opposed, by the coherence of 
the materia], to the sliding of one square inch upon the sur- 
face of another,* then will 7K sec. a represent the resistance 
which is overcome in the rupture of the column, so long as 
its height lies between the supposed limits ; which resist- 
ance being constant, the pressure applied upon the summit 
of the column to overcome it must evidently be constant. 
* Let this pressure be represented by P, and let CD 
p be the plane of rupture, ^ow it is evident that 
the inclination of the direction of P to the perpen- 
dicular QH to the surface of the plane, or its 
equal, the inclination a of CD to the base of the 
column, must be greater than the limiting angle 
of resistance of the surfaces ; if it were not, then 
would no pressure applied in the direction of P 



X 



be sufficient to cause the one surface to slide upon the other, 
even if a separation of the surfaces were produced along 
that plane. 

Let P be resolved into two other pressures, whose direc- 
tions are perpendicular and parallel to the plane of rupture ; 
the former will be represented by P cos. a, and the friction 
resulting from it by P cos. a tan. <p ; and the latter, repre- 
sented by P sin. a, will, when rupture is about to take place, 
be precisely equal to the coherence K7 sec. a of the plane of 
rupture increased by its friction P cos. a tan. <p, or P sin. 
a = Ky sec. a + P cos. a tan. 9, whence by reduction 

-p_ Ky cos. 9 2K7 cos. 9 . 6 



sin. (a— 9) cos. a sin. (2a-— 9) — sin. 9 

It is evident from this expression that it* the coherence of 
the material were the same in all directions, or if the unit of 

* The force necessary to overcome ;t resistance, such as thai here spoken 
of, has been appropriately called by Mr. Elodgkinson the force uecessary u 
shear it aero 



530 THl VI ANK of EUPTTBE. 

opposed to the slidingof on a of the 

masfi apon another were accurately the same in every di 
rion in which the plane CD may be ' s I to inn : 
the mass, then would the plane of actual rupture be incli 
to the r an angle represented by the formula 

'=1+1 (635); 

since the value of P would in this ease be (equation 634) 
a minimum when sin. (2a— 0) is a maximum, or when 

2a — z=-. or «=t+q : whence it follows that a plane in- 
clined to the base at that angle is that plane along which the 
rupture will first take place, aa P is gradually increased be- 
yond the limits of resistance. 

The actual inclination of the plane of rupture was found 
in the experiments of Mr. Hodgjkinson to vary with the ma- 
terial of the column. In cast iron, for instance, it varied 
g to the quality of the iron from 4 s " to 58°*, and 
different in different species. By this dependent 
the angle of rupture upon the nature of the material, it is 
proved that the value of the modulus of sliding cohe: 
z is not the same for every direction of the plane of rup- 
ture, or that the value of p varies greatly in different quali- 
ties of cast iron. 

Solving equation (634) in respect to 7 we obtain 

P 

> =y sin. (a— 0) cos. a sec. (636) ; 

from which expression the value of the modulus ;< may be 
determined in respect to any material whose limiting angle 
<»f resistance - is known, the force P producing rupture, 
under the circumstances supposed, being 1 . and alsc. 

the angle of rupture. f 



TlIK SECTION OF RUPTURE Of A BEAM. 

4"7. When a beam is deflected under a transverse strain, 

* Seventh Report of British Association, p. "49. 

•f A detailed statement of the ined in the experiments of Mr. 

Hodsrkinson on this subject is contained in the Appendix to the " Illustrations 
ut" Mechanics '" bv the author of this work. 



GENERAL CONDITIONS OF RUPTURE. 521 

the material on that side of it on which it sustains the strain 
is compressed, and the material on the opposite side 
extended. That imaginary surface which separates the 
compressed from the extended portion of the material is 
called its neutral surface (Art. 354.), and its position has 
been determined under all the ordinary circumstances of 
flexure. That which constitutes the strength of a beam is 
the resistance of its material to compression on the one side 
of its neutral surface, and to extension on the other ; so that 
if either of these yield the beam will be broken. 

The section of rupture is that transverse section of the 
beam about which, in its state bordering upon rupture, it is 
the most extended, if it be about to yield by the extension 
of its material, or the most compressed if about to yield by 
the compression of its material. 

In a prismatic beam, or a beam of uniform dimensions, it 
is evidently that section which passes through the point of 
greatest curvature of the neutral line, or the point in 
respect to which the radius of curvature of the neutral line 
is the least, or its reciprocal the greatest. 



General conditions of the rupture of a beam. 

408. Let PQ be the section of rupture in a beam sustain- 

^ p ing any given pressures, whose 

] 1 resultants are represented, if 

they be more in number than 

l 3{a three, by the three pressures P„ 

p ! ^0^^ ^ 2 ' ^ 3# ket the beam be upon 

A J^£2^..„ \ m the point of breaking by the 

i&zr. — r^5>^ I yielding of its material to exten- 

h , *\ § sion at the point of greatest ex- 

V ^3 tension P ; and let R represent, 

in the state of the beam border- 
ing upon rupture, the intersection of the neutral 9iirface 
with the section of rupture ; which intersection being in 
the case of rectangular beams a straight line, and being in 
fact the neutral axis, in that particular position which is 
assumed by it when the beam is brought into its state bor- 
dering upon rupture, may be called the axis of rupt^r* ; 
AK the area in square inches of any clement of the section 
of rupture, whose perpendicular distance from the axis of 
rupture R is represented by p ; S the resistance in pounds 



522 



GKNERAL CONDrriON8 OF Kt'PTURE 



opposed t<> the rapture of each square inch of the section at 
1*; c and <•„ the distances PR and QR in inches. 

The forces opposed per square inch to the extension and 
compression of the material at different points of the 
tion of rapture are t<> one another as their several perpen- 
dicnlar distances from the axis of rapture, if the elasticity 
of the material he supposed to remain perfect throughout 
the section of rupture, up to the period of rupture. 

Now at the distance c\ the force thus opposed to the 
extension of the material is represented per square inch by 
S ; at the distance p the elastic force opposed to the exten- 
sion or compression of the material (according as that 
distance is measured on the extended or compressed side), is 

therefore represented per square inch by — p, and the elastic 

force thus developed upon the element aK of the section of 

S 
rupture by — p^K, so that the moment of this elastic force 
c i 

S 
about R is represented by — p 2 -^K, and the sum of the mo- 

c \ 

ments of all the elastic forces upon the section of rupture 

S 
about the axis of rupture by — 2p 2 ^K ;* or representing the 

i 
moment of inertia of the section of rupture about the axis 
of rupture by I, the sum of the moments of the elastic 
forces upon the section of rupture about its axis of rupture 

SI 
is represented, at the instant of rupture, by — .f Now the 

elastic forces developed upon PQ are in equilibrium with 
the pressures applied to either of the portions APQD or 
BPQC, into which the beam is divided by that section; the 
sum of their moments about the point P is therefore equal 
to the moment of R, about that point. Representing, 
therefore, byjp, the perpendicular let fall from the point K 
upon the direction of P„ we have 



" It will be observed, as in Art. 358., that the elastic forces of extensiou 
ami those of compression tend to turn the surface of rupture in the same 
direction about the axis of rupture. 

\ This expression is tailed by the French writers the moment of rupture ; 
mi is of greater or less strength under given circumstances according 
a- it has a greater or less value. 



BY TBAJsSYEESE STEALS'. 523 

CT 

P^=- (637). 

409. If the deflexion be small in the state bordering upon 
rupture, and the directions of all the deflecting pressures be 
perpendicular to the surface of the beam, the axis of rupture 
passes through the centre of gravity of the section, and the 
value of c x is known. Where these conditions do not obtain, 
the value of c : might be determined by the principles laid 
down in Arts. 355. and 381. This determination would, 
however, leave the theory of the rupture of beams still in- 
complete in one important particular. The elasticity of the 
material has been supposed to remain perfect, at every point 
of the section of rupture, up to the instant when rupture is 
about to take place. I^ow it is to be observed, that by rea- 
son of its greater extension about the point P than at any 
other point of the section of rupture, the elastic limits are 
there passed before rupture takes place, and before they are 
attained at points nearer to the axis of rupture ; the forces 
opposed to the extension of the material cannot therefore be 
assumed to vary, at all points of PR, accurately as their dis- 
tances from the point R, in that state of the equilibrium of 
the beam which immediately precedes its rupture ; and the 
sum of their moments cannot therefore be assumed to be ac- 

SI 

curately represented by the expression — ■. This remark af- 

c i 

fects, moreover, the determination of the values of h and R 
(Arts. 355. and 381.), and therefore the value of c } 

To determine the influence upon the conditions of rupture 
by transverse strain of that unknown direction of the insistent 
pressures, and that variation from the law of perfect elasti- 
city which belongs to the state bordering upon rupture, we 
must fall back upon experiment. From this it has resulted, 
m respect to rectangular beams, that the error produced by 
these different causes in equation (637) will be corrected if 
a value be assigned to c x bearing, for each given material, a 
constant ratio to the distance of the point Pfrom the centre 
of gravity of the section of rupture ; so that c representing 
the depth of a rectangular beam, the error will bo corrected, 
in respect to a beam of any material, by assigning to <\ the 
value ///.]'', where m is a certain constant dependent upon 
the nature of the material. It i> evident that this cor- 
rection is equivalent to assuming '', = •''% and assigning 
to S the value -S instead of that which it has hitherto 



524 



[TCONS 01 BUPTUEE 



been supposed to represent, viz. the tenacity per square inch 
of the materia] of the beam. 

It is customary to make this assumption. The values of S 
corresponding t<> it have been determined, by experiment, 
in respect to the materials chiefly'used in construction, and 
will be found in a table at the end of this work. It is to 
these tables that the values represented by S in all subse- 
quent formulae are to be referred. 



410. From the remarks contained in the preceding article, 
it is not difficult to conceive the existence of some direct re- 
lation between the conditions of rupture by transverse and by 
longitudinal strain. Such a relation of the simplest kind ap- 
pears recently to have been discovered by the experiments 
of Mr. E. Eodgkinson*, extending to the conditions of rup- 
ture by compression, and common to all the different varie- 
ties of^naterial included under each of the following great 
divisions— timber, cast iron, stone, glass. 

The following tables contain the summary given by Mr. 
Hodffkinson of his results : — 



Description of Material. 


Assumed Crushing 

Strength per Square 

Inch. 


Mean Tensile 

Strength per Square 

Inch. 


Mean Transverse 

Strength of a Bar 

1 Inch Square and 

1 Foot Long. 


Timber - - - - 
Cast-iron ... 
Stone, including marble - 
Glass (plate and crown) - 


1000 

1000 
1000 
1000 


1900 
158 
100 
123 


85-1 
19-8 
9-8 
10" 



The following table shows the uniformity of this ratio in 
respect to different varieties of the same material : — 



Description of Material. 


Assumed Crushing 

Strength per Square 

Inch. 


Mean Tensile 
Strength per Squari 

Inch. 


Mean Transverse 

Strength of a Bar 

1 [neb Square and 

1 Foot Long. 


Black marble - 
Italian marble 
Rochdale flagstone- 
High Moorstone 
Yorkshire flag 
Stone from Little Hulton, 
near Bolton 


1000 
1000 
1000 

1000 
1000 

I 1000 


148 

84 
104 

100 

TO 


Krl 

Ketj 

9-9 

95 
8-8 



* This discovery was communicated to the British Association of Science at 
their meeting in 1 s J j ; it opens to us a new field of theoretical research. 



THE STRONGEST FORM OF SECTION. 525 



411. The strongest form of section at any given point 
fn the length of the beam. 

Since the extension and the compression of the material 
are the greatest at those points which are most distant from 
the neutral axes of the section, it is evident that the mate- 
rial cannot be in the state bordering upon rupture at every 
point of the section at the same instant (Art. 388.), unless all 
the material of the compressed side be collected at the same 
distance from the neutral axis, and likewise all the material 
of the extended side, or unless the material of the extended 
side and the material of the compressed side be respectively 
collected into two geometrical lines parallel to the neutral 
axis : a distribution manifestly impossible, since it would 
produce an entire separation of the two sides of the beam. 

The nearest practicable approach to this form of section is 
that represented in the accompanying figure, where the 
material is shown collected in two thin but wide flanges, 
united by a narrow rib. 

^ That which constitutes the strength of the 
beam being the resistance of its material to com- 
pression on the one side of its neutral axis, and 
its resistance to extension on the other side, it is 
evidently (Art. 388.) a second condition of the 
2j strongest form of any given section that when 
the beam is about to break across that section by 
extension on the one side, it may be about to break by com- 
pression on the other. So long, therefore, as the distribution 
of the material is not such as that the compressed and 
extended sides would yield together, the strongest form of 
section is not attained. Hence it is apparent that the 
strongest form of the section collects the greater quantity 
of the material on the compressed or the extended side of 
the beam, according as the resistance of the material to 
compression or to extension is the less. Where the material 
of the beam is cast iron", whose resistance to extension is 
greatly less than its resistance to compression, it Is evident 
that the greater portion of the material must be collected on 
the extended side. 

Thus, then, it follows, from the preceding condition and 

* It is only in cast iron beams that it is customary to seek an economy of 
the material in the strength of the section of the beam ; the Bame principle of 
economy i- Burely, however, applicable to beams of wood. 



520 



FORM OF SECTION. 



this, that the - form of section in «'i cast iron beam is 

that by which the material i> collected into two unequal 
flanges joined by a rib, the greater flange being on the 
extended side; and the proportion of this Inequality of the 
flanges being just such as to make up for the inequality of 
the resistances of the material to rupture by extension and 
compression respectively. 

Mr. Eodgkinson, to whom this suggestion is due, has 
directed a series of experiments to the determination of that 
proportion of the flanges by which the strongest form of 
section is obtained.-'' 

The details of these experiments are found in the following 
table :— 



Number of 
Experiment. 


Ratio of the Sections 
of the Flanges. 


Area of whole 

Section in Square 

Inches. 


Strength per Square 
Inch of Section in lbs. 


1 
2 
3 
4 
5 
6 


1 to 1- 
1 to 2- 

1 to 4' 
1 to 4-5 
1 to 5-5 
1 to 6-1 


2-82 

2-87 

3-02 

3-37 

5-0 

6-4 


2368 
2567 
2737 
3183 
3346 
4<>7.5 



In the first five experiments each beam broke by the tear- 
ing asunder of the lower flange. The distribution by which 
both were about to yield together — that is, the strongest 
distribution — was not therefore up to that period reached. 
At length, however, in the last experiment, the beam yielded 
by the compression of the upper flange. In this experiment, 
therefore, the upper flange was the weakest ; in the one be- 
fore it, the lower flange was the weakest. For a form 
between the two, therefore, the flanges were of equal strength 
to resist extension and compression respectively ; and this 
was the strongest form of section (Art. 388.). 

In this strongest form the lower flange had six times the 
material of the upper. It is represented in the accompany- 
ing figure. 

A B In the best form of cast iron beam or 

girder used before these experiments, 
there was never attained a strength of 
more than 2885 lbs. per square inch of 
ion. There was. therefore, by this 
form, a gain of 1190 lbs. per square inch 
ID of the section, or of fths the strength of 
the beam. 



* Memoirs of Manchester Philosophical Society, vol. 
tions of Mechanics, Art. 68. 



p. 453. Illustra- 



rnE b;:.vm of gskatest strength, 527 



412. The section of rupture. 

rupture being determined in respect to 
any section of the beam by equation (637), it is evident that 
the particular section across which rupture will actually take 
place is that in respect to which equation (637) is first satis- 
fied, as P, is continually increased ; or that section in respect 
to which the formula 

£ (638 ) 

is the least. 

If the beam be loaded along its whole length, arid x repre- 
sent the distance of any section from the extremity at which 
the load commences, and /x the load on each foot of the 
length, then (Art. 371.) Pj^ is represented by %px*. The 
section of riipture in this case is therefore that section in 
respect to which (*> is first made to satisfy the equation 

SI 
%iw?= — ; or in respect to which the formula 

k ^ 

is the least. 

If the section of the beam be uniform, — is constant ; the 

section of rupture is therefore evidently that which is most 
distant from the free extremity of the beam. 



4:13. The beam of greatest strength. 

The beam of greatest strength being that (Art. 388.) which 
presents an equal liability to rupture across every section, or 
in respect to which every section is brought into the state 
bordering upon rupture by the same deflecting pressure, is 
evidently that by which a given value of Pis made to satisfy 
equation (637) for all the possible values of I, ]?,, and o a or 
in respect to which the formula 



— (640) 

PA v ; 



ib constant. 






TUE B'J LEN< NI OF BEAMS. 



[f the beam be uniformly loaded throughout (Art. 371.), 
this condition becom 



_I_ 



(641), 



or constant, for all points in the length of the beam. 



414. One extremity of a beam is firmly imbedded in 
masonry, and .v pressure is applied to the othek 
extremity in a direction i'kitpendicular to its length: 
to determine the conditions of the rupture. 

If x represent the distance of any section of the beam 
from the extremity A to which the load P 
is applied, and a its whole length, and if the 
section of the beam be everywhere the 
same, then the formula ( 638 ) is least 
at the point B, where x is greatest : at 
this point, therefore, the rupture of the 
beam will take place. Representing by 
P the pressure necessary to break the 
beam, and observing that in this case the 
perpendicular upon the direction of P 
from the section of rupture is represented 
by '■', we have (equation 637) 

ST 

P = - (642). 






u 

I , 1 ' I ' I 



5 



If the section of the beam be a rec- 
tangle, whose breadth is b and its depth c, 
then I= T \bc\ 0,=^. 



--.p=*s 



be' 



(643). 



If the beam be a solid cylinder, whose radius is <?, then 
(Art. 364.) I=i«c\ <?,=<?. 



•p=i-s- 



(644). 



If the beam be a hollow cylinder, whose radii are r t and 
r 2 , I = J-( , /' 1 4 — /•.,*» ; which expression may be put under the 
form -'•//•" ■ Art. S6.), '/'representing the meaD 



THE STRENGTH OF BEAMS. 



529 



radius of the hollow cylinder, and c its thickness. Also 
c x —i\—r+\c\ 



:.P=«rS- 



(r+ic)a 



(645). 



415. The strongest form of beam under the conditions sup- 
posed in the last article. 



1st. Let the section of the beam be a 
rectangle, and let y be the depth of 
this rectangle at a point whose distance 
from its extremity A is represented by 
a?, and let its breadth b be the same 
throughout. In this case I=xV^y 8 > 
c x -=^y\ therefore (equation 637) P= 

QT 2 

— =±Sb— . If, therefore, P be taken 
c x x x 

to represent the pressure which the- 
beam is destined just to support, then 
the form of its section ABC is deter- 
mined (Art. 413.) by the equation 

. 6P 




y 



Sb 



(646); 



it is therefore a parabola, whose vertex; 
is at A.* 

If the portion DC of the beam do not rest against 
a masonry at every point, but only at its 

extremity D, its form should evidently be 
the same with that of ABC. 

2d. Let the section be a circle, and 
let y represent its radius at distance x 
from its extremity A, then I=-}*y\ 

y % 

c x —y\ therefore P= J*S - so that the 

geometrical form of its longitudinal 
section is determined by the equa- 
tion 




* The portion of the beam imbedded in the masonry should have tho form 
described in Art. 417. 

34 



530 



TIIIC STRENGTH OF BEAMS. 



, 4P 

V= Vif 



(647), 



r representing the greatest pressure to which it is destined 
to be subjected. 



416. Tjie conditions of the rupture of a beam supported 
at one extremity, and loaded throughout it8 whole 

LENGTH. 

Representing the weight resting upon each inch of its 

length a by i^, and observ- 
ing that the moment of the 
weight upon a length x of 
the beam from A, about the 
corresponding neutral axis, 
is represented (Art. 371.) 
by -J-M-a? 2 , it is apparent (Art. 
412.) that, if the beam be 
of uniform dimensions, its 
section of rupture is BD. 
Its strength is determined 
by substituting ^a? for P,^>, 
in equation (637), and solving in respect to m- ; we thus obtain 

V-=^k (6«); 




by which equation is determined the uniform load to which 
the beam may be subjected, on each inch of its length. 

For a rectangular beam, whose width is b and its depth 
c, this expression becomes 



Sfo a 
3a* 



. (649). 



417. To determine the form of greatest strength (Art. 413.) 
in the case of a beam having a rectangular section of uni- 
form breadth, \?-< ? must be substituted for F,p, in equation 
(637), and &hf for I, and \y for e l ; whence we obtain by 
reduction 



3mV 



(S) 



(650. 



THE STRENGTH OF BEAMS. 



531 




The form of greatest strength is therefore, in this case, the 
straight line joining the points A and B ; the distance DB 
being determined by substituting the distance AD for x in 
the above equation. 

That portion BED of the beam which is embedded in the 
masonry should evidently be of the same form with DBA.* 



418. If, in addition to the uniform load upon the beam, a 
given weight W be suspended from A, -^a? a + W# must be 




substituted for Pj?, in equation (637) ; we shall thus obtain 
tor the equation to the form of greatest strength 



V 



3f* j 2W , t 



(651), 



which is the equation to an hyperbola having its vertex 
at A.f 



* It is obvious that in all cases the strength of a beam at each point of ita 
length is dependent upon the dimensions of its cross section at that point, and 
tnat its general form may in any way be changed without impairing its strength 
provided those dimensions of the section be everywhere preserved. 

f Church's Anal. Geom. Art. 124. 



532 



THE STRENGTH OF BEAMS. 



4 1!'. Tbb BEAM of greatest strength in reference to the 

Km KM OF ITS SECTION AND TO THE VARIATION OF THE 

ONE 
UNI- 



KXTBEMTTT IN A HORIZONTAL POSITION, AND LOADED 
FORMLT THROUGHOUT ITS LENGTH. 




The general form of the section must evidently be that 

described in Art. 411. Let 
the same notation be taken 
as in Art. 365., except that 
the depth MQ of the plate 
or rib joining the two 
flanges is to be represented 
by y, and its thickness by <?, 
so that d 3 =y, and A 3 =cy ; 
therefore by equation (503), 

Also representing by c l the distance of the centre of gravity 
of the whole section from the upper surface of the beam, 
we have _ cjjk, + A 2 + cy)= {j& + d n )cy + (y + d, + M) A l + id 9 
A 2 . Substituting for I and c x in equation (637), and for P,j9, its 
value •J-M-a? 3 , x being taken to represent the distance AM, and 
P the load on each inch of that length, we have (Art. 
413.) 

3,a 2 _ 

( A l d 1 > + A,d; + cy>)(A i + A a + cy )+\ l2A l A, + S(A 1 +^)cy\ y> 

(652). 

Let the area cy of the section of the rib now be neglected, 
as exceedingly small when compared with the areas of the 
sections of the flanges, an hypothesis which assigns to the 
beam somewhat less than its actual strength; let also the 
area of the section of the upper flange be assumed equal to 
// times that of the lower, or A U =?*A„ 

# 3M4B" • (n+l)W+nd/)+12n^ 



"SA. 



(*+*)+«*+»)* (G53) - 

be exceedingly thin, d x and d q are exceed- 
The equation will then 



If the flanges 

ingly small and may be neglected. 



THE STRENGTH OF BEAMS. 



533 



become that to a parabola whose vertex is at A and its axis 
vertical. This may therefore be assumed as a near approxi- 
mation to the true form of the curve AQC. 

Where the material is cast iron, it appears by Mr. Hodg- 
kinson's experiments (Art. 411.) that n is to be taken=6. 



420. A BEAM OF UNIFORM SECTION IS SUPPORTED AT ITS 
EXTREMITIES AND LOADED AT ANY POINT BETWEEN THEM I 
IT IS REQUIRED TO DETERMINE THE CONDITIONS OF RUPTURE. 

The point of rupture in the case of a uniform section 

is evidently (Art. 412.) the point 
C, from which the load is sus- 
pended; representing AB, AG, 
BC, by a, &„ and & 2 ; and ob- 
serving that the pressure P, 

Wa 

upon the point B of the beam = -, so that the moment 




of P 1? in respect to the section of rupture C = 



Wa,a n 



a 



we 



have, by equation (637). 



\W 



a 

Sla 

a Y a^c 1 



SI 



(654). 



(655); 



If the beam be rectangular, I=^bc 9 , c 1 =^ic, 

i¥= sw« 

6 a x a 2 ' 

where W represents the breaking weight, S the modulus of 
rupture, a the length, b the breadth, c the depth, and a„ a, 
the distances of the point C from the two extremities, all 
these dimensions being in inches. 

If the load be suspended in the middle, a t =a 2 =ia, 

'•»-?? <«>• 

If the beam be a solid cylinder, whose radius =<?, then 1 = 
frrc\ c x —c\ therefore, equation (654), 

W=^^! ( 6 57). 

4 a,a 2 



534 Tin: .11 of beam-. 

It* the beam be a hollow cylinder, whose mean radius 
and its thickness c, I=w(f*+iflfy ^, = /■ + £c* ; therefore, 
equation (654), 

W =^«^±gl (658). 

If tlie Bection of the beam be that represented in Art. 411., 
being everywhere of the same dimensions, then, observing 
that A',^ V A ' A., nearly, we have, (equations 503 and 
654) 

W= 6 (2A 1 +A,)a 1 a 2 rf, * ' ' < 659 >- 

where A,, A 3 represent the areas of the sections of the upper 
and lower flanges, and A, that of the connecting rib or plate, 
and d„ d„ d % their respective depths. 



421. A BEAM IS SUPPORTED AT ITS EXTREMITIES, AND LOADED 
AT ANY GIVEN POLNT BETWEEN THEM \ ITS SECTION IS OF A 
GP7EN GEOMETRICAL FORM, BUT OF VARIABLE DLMENSI<>N- \ 
IT IS REQULRED TO DETERMINE THE LAW OF THIS VARIA- 
TION, SO THAT THE STRENGTH OF THE BEAM MAY BE A 
MAXIMUM. 

W representing the breaking load upon the beam, and 

<z„ a a the distances of its point 
of suspension C, from A and 
B, the pressure P, upon A is 

represented by a . If, there- 
fore (Art. 388.), x represent 

the horizontal distance of any section MQ from the point of 
support A. and 1 its moment of inertia, and c, the distance 
from its centre of gravity to the point where rupture is about 
to take place (in this case its lowest point) ; then by equa- 
tion (63 7 j 

tH? « 

1st. Let the section be rectangular : let its breadth b be 
constant ; and let its depth at the distance x from A e 




THE STRENGTH OF BEAMS. 535 

represented by y ; therefore I= T \by% c 1 =iy. SLbstituting 
in the above equation and reducing, 

HeP- (661 > 

The curve AC is therefore a parabola, whose vertex is at 
A, and its axis horizontal. In like manner the curve BC is 
a parabola, whose equation is identical with the above, ex- 
cept that a x is to be substituted in it for a 2 . 

2d. Let the section of the beam be a circle. Represent- 
ing the radius of a section at distance x from A by y, we 
have I^J^y 4 , e l =y, therefore by equation (660) 

*=%% . . . . , (662). 

3d. Let the section of the beam be circular ; but let it be 
hollow, the thickness of its material being every where the 
same, and represented by c. If y= mean radius of cylinder 
at distance x from A, then I^tcy^f + ic), c l —{y J t^c) ; 

■■■*=mw <"'• 



422. The beam of greatest absolute strength when 
loaded at a grven point and supported at the extre- 
MITIES. 

Let the section of the beam be that of greatest strength 
(Art. 411.). Substituting in equation (660) the value of — 
as before in equation (652), and reducing, 

6Wa 2a; = (A 1 ^ 2 +A 2 ri 2 a + C i / 3 )(A 1 +A 2 +c.y)+12A 1 A 2 ,y ; '4-3(A 1 +A 2 )ci/' 

Sa (y+2*)cy+2(y-|^+Jrfi)Ai+A a * ..(604). 

If the section cy of the rib be every where exceedingly 
small as compared with the sections of the flanges, and if 
A 2 =nA„ 

ma (n + 1) W+nd\) + 12ny> 

SA x a ' 2y + d 1 + (n + ZyI^~ ^°°> 

There is a value of a? in this equation for which y becomes 



536 THE STRENGTH OF BEAMS. 

impossible. For value- Less than this, the condition of uni 
form Btrength cannol therefore obtain. It is only in respect 
to those parts of the beam which lie between the values of 
r (measured from the two points of Bupport) for which y 
thus becomes impossible, that the condition of greatest 
st length (Art. 388.) is possible. If its proper value be 
assigned to n (Art. 411.), this may be assumed as an approxi- 
mation to the true form of beam of the greatest absolute 
strength". When the material is cast iron, it appears by the 
experiments of Mr. Hodgkinson (Art. 411.) that ?i=6. A, 
represents in all the above cases the section of the extended 
flange ; in this case, therefore, it represents the section of 
the lower flange. 

The depth CD at the point of suspension may be deter- 
mined by substituting a x for x in equation (665)', its value is 
thus found to be represented by the formula 

CD~^ (666). 



423. If instead of the depth of the beam being made to 
vary so as to adapt itself to the condition (Art. 388.) of uni- 
form strength, its breadth b be made thus to vary, the depth 
c remaining the same ; then, assuming the breadth of the 
upper flange at the distance x from the point of support A 
to be represented by y, and the section of the lower flange 
to be n times greater than that of the upper; observing, 
moreover, that in equation (503) A 1 =yd 1 , A^=nA i =nyd 1 ; 
neglecting also A 3 as exceedingly small when compared with 
A, and A 2 , and writing c for d 3 , we have by reduction, 

71+1 

Also c, being the distance of the lower surface of the beam 
from the common centre of gravity of the sections of the 
two flanges, we have c^n + lj^c. Eliminating, therefore, 
the values of I and c, from equation (660), 

•HSl, i aoh- 1 ) w+«*)§ + ned > \y ( 6G7 )> 

the equation to a straight line. Each flange is therefore in 
this case a quadrilateral figure, whose dimensions are deter- 
mined from the greatest breadth; this last being known, foi 



THE STRENGTH OF BEAMS. 



537 



the upper flange, by substituting a x for x in the above equa- 
tion, and solving in respect of y, and for the lower flange 
from the equation nb 1 d 1 =h 2 d^ in which &,, h 2 represent the 
greatest breadths of the two flanges, and d x , d 2 their depths 



424. A BEAM IS LOADED UNIFORMLY THROUGHOUT ITS WHOLfl 
LENGTH, AND SUPPORTED AT ITS EXTREMITIES : IT IS REQUIRED 
TO DETERMINE, 1. The CONDITIONS OF ITS RUPTURE WHEN ITS 
CROSS SECTION IS UNIFORM THROUGHOUT ; 2. The STRONGEST 

form of beam having every where a rectangular cross 
section ; 3. The beam of greatest strength in refer- 
ence BOTH TO THE FORM AND THE VARIATION OF ITS CROSS 
SECTION. 

1. If the section of the beam be uniform, its point of rup- 
ture is determined by formula (639) 
to be its middle point. Representing, 
therefore, in this case, the length of 
the beam by 2a, the weight on each 
inch of its length by v<, and its breadth 
by h ; and observing that in this case 
P^zz^-f^^i^^we have by 
equation (637) 




{j.— 



2S1 
tfc x 



(668), 



where ^ represents the load per inch of the length of the 
beam necessary to produce rupture. In the case of a rectan- 
gular beam, this equation becomes 

'=8? (669 > 



1 1 




m \ .. ;, . 




-!' 1 


^^/~ 


T7T- 


1 




D 


<i 




-j- 






1 


1 






r 


1 







2. To determine the form of the beam of greatest strength 
having a rectangular section of 
given breadth 5, let y be taken to 
represent its depth PQ at a point I', 
and x its horizontal distance from 
the point A. Then I = ,' h,f. 
c x —\y\ also Vjj, (equation 637) 
representing the moment of the resultanl of tne pressures 
upon AP about the centre of gravity of PQ=f*aa>— 1 
therefore by equation (637) pax— i^^Sbf ; 



538 



THE STRENGTH OF BEAMS. 



the equation to an ellipse, whose vertex is in A, and ite 
centre at C. 

3. To determine the beam of absolute maximum strength, 
let it bo assumed, as in Art. 422, that the area of the section 
of the rib is exceedingly small as compared with the areas 
of the sections of the flanges ; and let the area of the section 
of the lower or extended flange be n times that of the upper ; 

{?i + l)(d;+nd;) + 12?iy* ) 
2i/ + d 1 + {n + 2)d 7 ) 

also Yj) 1 =^ax— l^a? 2 ; whence, by equation (637), 



then, as in Art. 422 



22 l-± | 

' c~ 6 ( 



uax— i^x' i = 



SA, 



(n + l)(tf l ' + ttO + 12;iy> ) 




4. If it be proposed to make the rib or plate uniting the 
two flanges everywhere of the same depth,* and so to vary 
the breadths of the flanges as to give to the beam a uniform 
strength at all points under these circumstances ; represent- 
ing by y the breadth of the upper flange at a horizontal 
distance x from the point of support, we shall obtain, as iu 
Art. 423, 



~cr y 



(n+1 ) 

1'IC 



(d* +nd?)d l + yncd l . 



Moreover, T l p l =^ax— ^x 2 =^x(2a— x) ; whence we obtain 
by substitution in equation (637), and reduction, 

x{2a-x)= (H) j(»+l) W+nd,*)+12nJ\y (673) ; 

the equation to a parobola,f whose axis is in the horizontal 
line bisecting the flange at right angles, its parameter repre 



* As in Mr. Uodgkinson's construction, 
f Church's Anal. Geom. Art. 171. 



THE STRENGTH OF BEAMS. 53Q 

sented by the coefficient of ym the preceding equation, and 
half the breadth ot the flange in the middle determined by 
the formula J 

6 c«V 

\(n + 1; [ci; + nd- 2 ') + \2n&\ Sd, ( 6U )' 

The equation to the lower flange is determined by substi- 
tuting for y, in equation (673), g ; whence it follows that 
the breadth of the lower flange in 'the middle is equal to 

that of the upper multiplied bj the fraction — 2 

nd x ' 

425. A RECTANGULAR BEAM OF UNIFORM SECTION, AND UNI- 
FORMLY LOADED THROUGHOUT ITS LENGTH, IS SUPPORTED BY 
TWO PROPS PLACED AT EQUAL DISTANCES FROM ITS EXTREM- 
ITLES: TO DETERMINE THE CONDITIONS OF RUPTURE. 

It is evident from formula (639) that the section of rup- 
,.;■;■■■■■'.'■■,■.■.■■■■■■•_ ■ _■ tur e of the portion CA of the 
■ : ■ : ■ ' : I ■ : : : ' I ■ . ■ : - , ■ . | ■ I : : beam is at A, and therefore that 
—~^^-~^m^j^ V 16 collditions of its rupture are 
o determined (Art. 416.) by the 
equation 

^=8? ( 675 )5 

lft fl H „nAn i ■ i J* 1 ^™ ?> represents, as before, the 
load upon each inch of the length of the beam b its 
breadth, .its depth, and a x the length of the portion AC 

tio^AR' nA eV t nt that tlie P° int of mpture of the por- 
tion AB of the beam is at E. Now the value of P » 
(equation 637) is, in respect to the portion AE of the beam! 
^{a-aj-^a ; 2a representing the whole length of the 
beam a the load upon each inch & of the length Ae beam 
which would produce rupture at E, and therefore m the 
resistance ot each prop in the state bordering upon rupture ; 

also -=ibc\ Whence, by equation (637), M*-<*0- 



S^ a 



540 T!I! I-: N ; I II OF BEAM8. 



420. The best positions of the props. 

If the load (* be imagined to be continually increased, it 
is evidenl that rupture will eventually take place at A or at 
E according as the Limit represented by equation (675), or 
that represented by equation (676), is first attained, or 
according as jx, or m* 9 is the less. 

Let [*, be conceived to be tne less, and let the prop A be 
moved nearer to the extremity C; a l being thus diminished, 
f*, will be increased, and >j-. 2 diminished. Now if, after this 
change in the position of the prop, |*, still remains less than 
|x 2 , it is evident that the beam will bear a greater load than 
it would before, and that when by continually increasing 
the load it is brought into the state bordering upon rupture 
at A it will not be in the state bordering upon rupture at E. 
The beam may therefore be strengthened yet further by 
moving the prop A towards C ; and thus continually, so 
that the beam evidently becomes the strongest when the 
prop is moved into such a position that ^ may just equal 
f* 9 . This position is readily determined from equations (675) 
and (676) to be that in which 

a=a( 4/2~-l)=-414213oa (677). 



427. A RECTANGULAR BEAM OF UNIFORM SECTION AND UNI- 
FORMLY LOADED IS SUPPORTED AT ITS EXTRKMITIES, AND BY 
TWO PROPS SITUATED AT EQUAL DISTANCES FROM THEM I TO 
DETERMINE THE CONDITIONS OF RUPTURE. 

Adopting the same notation as in Art. 374., it appears by 

equation (543) that the dis- 
tance x l of the point of great- 
est curvature of the neutral 
line, and therefore of the sec- 
tion of rupture in AB from 
A (Art. 407.) being that 

oVy ■ . , 
where -~* is the greatest, is 

determined by the equation 

* The curvature of the neutral line being everywhere exceedingly small, 

— mav be assumed =1. Tho expression for the radius of curvature in terms 

ax 

of tho rectangular co-ordinates resolves itself therefore, in this case, into th« 

Becond differential coefficient. 



; ■ ; ■ ; i ; ■ ; i ; i ; , ; , ; r-^ 




THE STRENGTH OF BEAMS. 



541 



f*a? t =P, it being observed that, at the section of rupture, the 
neutral line is concave to the axis of x, and therefore the 
second differential coefficient (equation 543) negative. The 
value of P is that determined by equation (551) ;. so that 

, ^ 3 + 12^ 2 - 24^ + 8 
">=**• n{2n-S) ^ 678 )' 

where a represents the distance AE, and na the distance AB. 
Let P represent the intersection of the neutral line with 
the plane of rupture, and p 1 the load per inch of the whole 
length of the beam which would produce a rupture at P. 
Now the sum of the moments of the forces impressed on 
AP (other than the elastic forces on the section of rupture) 
is represented in the state bordering upon rupture, by 

P^— a^i 2 ; or, since P 1 =f* 1 » 1J it is represented by ^— P x *; 

whence it follows by equation (637) that the conditions of 
the rupture of the beam between A and 33 are determined 

by the equation^— P 1 a =}S5c a , or, 



P^^SW 



(679). 



Eliminating the value of P a between equations (551) and 
(679), we obtain 



Mv 



Sfo a ( 

'So? \n 3 + l2n 



$n(Zn—S) 



24^ + 8 



(680). 



Substituting this value of f\ in equation (679), and 
8S£c 2 i n(2n-S) \ 



reducing 



P,=- 



Za 



n 



\W— 24/1 + 8 



(681). 




fV 



; Tu? 



If the points B and C 
coincide, or the beam be 
supported by a single prop 
in the middle, n=l; there- 
fore, by equations (680") and 
(681), 



(682); 



542 TIIE STRENGTH OF BEAMS. 

P --9^ ^ 

Similarly, it appears by equation (54:7) that the point < f 
greatest curvature between B and C is E ; if the rupture of 
the beam take place first between these points, it will there- 
fore take place in the middle. Let f< 2 represent the load, 
per inch of the length, which would produce a rupture at E. 
Now, the sum of the moments about E of the forces im- 
pressed upon AE is l > l a + ¥ 1l (a—na)—^a?=(J > x + Y^a— 
r i na—i^a*=^a*—(^a—F i )na—i^a i (since P,+ P 3 =fx a a)= 
i(l—2n)'^ i a 1 -\-i > 1 na. Therefore by equation (637) 

i(l-2^)^ + P 1 ^=iSk 2 (684). 

Substituting for P x its value from equation (551), and 
solving in respect to i^ 2 , 

4 S6c 2 ( 271-3 ) , flQK , 

If the load be continually increased, the beam will break 
between A and B, or between B and C, according as f* - 
(equation 680) or ^ 3 (equation 685) is the 



428. The best positions of the trops. 

It may be shown, as in Art. 426., that the positions in 
which the props must be placed so as to cause the beam to 
bear the greatest possible load distributed uniformly over its 
whole length, are those by which the values of p 1 (equation 
680) and f* a (equation 685) are made equal ; the former of 
these quantities representing the load per inch of the length, 
which being uniformly distributed over the whole beam 
would just produce rupture between A and B, if it did not 
before take place between B and C ; and the latter that 
which would, under the same circumstances, produce rup- 
ture between B and C if it had not before taken place 
between A and B. 

Let, then, na represent the distance at which the prop B 
must be placed from A to produce this equality ; and let the 
value of M-, given by equation (679) be substituted for ^ in 
equation (6S4) ; we shall thus obtain by reduction 

p Sbc'n (SbcJ 

1 + ' ' 3(l-2n)-9(l-2n) 
Solving this quadratic in respect to P,<z, 



THE STRENGTH OF BEAMS. 543 

\ l-2n )• 

The negative sign must be taken in this expression, since 
the positive would give P 1=(V , by equation 679), and cor- 
responds therefore to the case nJo. Assuming the negative 
sign, and reducing, we have 3(2n~1)P ia =Sbl Sabftitut- 

ant reducing^ 881011 *" *' ^ ^ ^ ^ Uati0n « 

8n(2^-l) (2^-3) _ 

•oSl^Th fi 0< ? of ^ is , e( l uat . i ^ ^ 1-57087, '61078, and 
^6994. The first and last are inadmissible ; the one carry- 
mg the point B beyond E, and the other aiignWtoPa 
negative value.* The best position of the prop s fherefore 
that which is determined by the value P meretore 

n= -61078 (686). 



LZ T C ° NDITI0NS ° F ™ E R ^URE OF A RECTANGULAR 
BEAM LOADED UNIFORMLY THROUGHOUT ITS LENGTH, AND 
HAVING ITS EXTREMITIES PROLONGED AND FIRMLY EMBEDDED 
IN MASONRY. 

It has been shown (Art. 376.) that the conditions of the 
deflexion of the beam are, in this case, the same as though 
its eternities having been prolonged to a point A (see fig. 

Lr^'h that A ^ mi S ht «l« al '6202AE, had been sup- 

ported by a prop at B, and by the resistance of any fixed 
7tlT A ' ^l^whicfi would produce the rupture 
ot the beam is therefore, m this case, the same as that which 

Mv te w * r T ,U ' e ° f a beam s »PPOrted by props 
(Ait. 427.) between the props, and is determined by that 
value of>, (equation 685) which is given by the value -6202 
oi n. it is, however, to be observed that the symbol a 

™rt^ e f ma - v ; neTer ,"' cIess ' S "PP° SC the extremity A, instead of beinc ran 

Cm ah™"' tZ c' '° ,>0 , ' i " ned <l0 "" b ' V » B ** i " '"' ' I'"-™" «* Z 
nmn. may occur in practice, and the besl posil ,1 the 

SSK^Kxa.* " is tl,ilt lv " k '" is toCT «™«> "y the ,,':,-, ,. :,!■ !!:; 



AA 



THE STRENGTH OF COLUMNS. 



s 



r~r-^-r 



nzzr 



T — r 



- ,x_ . ; 



j i i 



represents in that equation 
the distance AE (Jig. Art. 
427.) ; and that if we take 
it to represent the distance 

BE in that or the accompa- 
nying figure, we must snb- 
a 



stitute 



1-71 



for a in equa- 



tion (685), since a=BE=AE— AI3=(1— n)AE ; so that 
AE=- . This substitution being made, equation (685), 



becomes 



ft 



4 S^ (2n-S)(l-7iy 
3 a » n »_4(i_ n y 



and substituting the value *6202 for n, we obtain by reduc- 
tion 

Sfo* faQfr . 

^■=-^r ( 68 0> 

by which formula the load per inch of the length of the 
beam necessary to produce rupture is determined. 

If the beam had not been prolonged beyond the points of 
support B and C and imbedded in the masonry, then the 
load per inch of the length necessary to produce rupture 
would have been represented by equation (669) : eliminat- 
ing between that equation and equation (687), we obtain 
^=3** ; so that the load per inch of the length necessary to 
produce rupture is 3 times as great, when the extremities 
of the beam are prolonged and firmly imbedded in the ma- 
sonry, as when they are free ; i. e. the strength of the beam 
is 3 times as great in the one ease as in the other. 



430. The strengh of columns. 



For all the knowledge of this subject on which any reli- 
ance can be placed by the engineer he is endebted to expe- 
riment.* 

* The hypothesis upon which it has been customary to found the theoretical 
discussion of it, is so obviously insufficient, and the results have been shown 
by Mr. Bodgkinson to be so little in accordance with those of practice, that 
the high sanction it has received from labours such as those of Euler, Lagrange, 
Poissou, and Navier, can no longer establish for it a claim to be admitted 
among the conclusions of science. (See Appendix K.) 



THE STRENGTH OF COLUMNS. 



545 



The following are the principal results obtained in the 
valuable series of experimental inquiries recently instituted 
by Mr. Eaton Hodgkinson.* 

Formulae representing the absolute strength of a cyl- 
indrical COLUMN TO SUSTAIN A PRESSURE IN THE DIRECTION 
OF ITS LENGTH. 

D= external diameter or side of the square of the column 
in inches. 

D 1 =internal diameter of hollow cylinder in inches. 
L= length in feet. 
W=breaking weight in tons. 



Nature of the Column. 



Solid cylindrical 
cast iron - 



column of ) 



Hollow cylindrical column of 

cast iron - 
Solid cylindrical column of 

wrought iron 

Solid square pillar of Dantzic 

oak (dry) - 
Solid square pillar of red deal 

(dry) 



Both Ends being 

rounded, the Length of 

the Column exceeding 

fifteen times its 

Diameter. 



W=14-9- 



W=13 



D'-^-Df 



W=42-8 : 



L 1 ' 7 

D 3.76 

L 5 " 



Both Ends being flat,. the 

Length of the Column 

exceeding thirty times its 

Diameter. 



W=44-16 
W=44-34 



T) 3 . 66 

D 3.55_ Bi 3.65 



W=133-75 



D a 



D 4 

W=10-95-^ 
L 

D* 



In all cases the strength of a column, one of whose ends 
was rounded and the other flat, was found to be an arith- 
metic mean between the strengths of two other columns of 
the same dimensions, one having both ends rounded and the 
other having both ends flat. 

The above results only apply to the case in which the 
length of the column is so great that its fracture is produced 
wholly by the lending of its material ; this limit is fixed by 
Mr. Iloclgkinson in respect to columns of cast iron at about 
fifteen times the diameter when the extremities are rounded, 

* From a paper by Mr. ITodgkinson, published in the second part of the 
Transactions of the Royal Society for 1840, to which the royal medal of 1 1 1 * - 
Society was awarded. The experiments were made at (he expense of 
Mr. Fairbairn of Manchester, by whose liberal encouragement the researches 
of practical science have been in other respects so greatly advanced. 

35 



546 T0R8I0N. 

and thirty times the diameter when they are flat. In 
shorter columns fracture takes place partly by the crushing 
and partly by the bending of the material. To these shorter 
columns the following rule was found to apply with suf- 
ficient accuracy: — ki W W, represent the weight in tons 
which would break the column by bending alone (or if it 
did not crush) as given by the preceding formula, and W 2 
the weight in tons which would break the column by crush- 
ing alone (or if it did not bend) as determined from the 
preceding table, then the actual breaking weight W of the 
column is represented in tons by the formula 

WW 

w =WTfw; W- 

Columns enlarged in the middle. It was found that the 
strengths of columns of cast iron, whose diameters were from 
one and a half times to twice as great in the middle as at 
the extremities, were stronger by one seventh than solid 
columns, containing the same quantity of iron and of the 
same length, when their extremities were rounded ; and 
stronger by one eighth or one ninth when their extremities 
were flat and rendered immoveable by discs. 



431. Relative strength of long columns of cast iro.n, 
wrought iron, steel, and timber of the same dimensions. 
— Calling the strength of the cast iron column 1000, the 
strength of the wrought iron column will, according to these 
experiments, be 1745, that of the cast steel column 2518, of 
the column of Dantzic oak 108*8, and of the column of red 
deal 78-5. 

Effect of drying on the strength of columns of timber. — 
It results from these experiments, that the strength of short 
columns of wet timber to resist crushing is not one half 'that 
of columns of the same dimensions of dry timber. 



Torsion. 
432. The elasticity of torsion. 
Let ABCD represent a solid cylinder, one of whose trans- 



TORSION 



54:7 




verse sections AEB is immoveably fixed, 
and every other displaced in its own plane, 
about its centre, by the action of a pres- 
sure P applied, at a given distance a from 
the axis, to the section CD of the cylinder 
in the plane of that section and round its 
centre ; the cylinder is said, under these 
circumstances, to be subjected to torsion* 
and the forces opposed to the alteration of 
its form, and to its rupture, constitute its 
resistance to torsion. 

Let aahfi be any section of the cylinder 
whose distance from the section AEB is 
represented by x, and let a/3 represent that 
diameter of the section auhfi which was 
parallel to the diameter AB before the torsion commenced ; 
let ab be the projection of the diameter AB upon the sec- 
tion aubfi, and let the angle aca be represented by Q. 

Now the elastic forces called into action upon the section 
aabfi are in equilibrium with the pressure P. But these 
elastic forces result from the displacement of the section 
aabfi upon its immediately subjacent section. Moreover, 
the actual displacement of any small element aK of the 
section aoh$, upon the subjacent section, evidently depends 
partly upon the angular displacement of the one section 
upon the other, and partly upon the distance p of the 
element in question from the axis of the cylinder. Now the 
angle oca, or & is evidently the sum of the angular displace- 
ments of all the sections between aocbfi and AEB upon their 
subjacent sections ; and the angular displacement of each 
upon its subjacent section is the same, the circumstances 
affecting the displacement of each being obviously the same : 
also the number of these sections varies as x, and the sum 
of their angular displacements is represented by d ; there- 
fore the angular displacement of each section upon its sub- 

jacent section varies as -, and the actual displacement of 



the small element aK of the section aabfi varies as -p. 

x v 



Now 



the material being elastic, the pressure which must be 
applied to this element in order to keep it in this Btate of 
displacement varies as the amount or the displacement 



(Art. 345.), or as -p. Let its actual amount, when referred 
x 



548 TORSION. 



to a anil of Burface, be represented by G-p, where G is a 

x 

ant dependant for its amount on the elastic; 
qualities of the material, and called the modulus of torsion ; 
then will the force of torsion required to keep the element 

& 
Mi in its state of displacement be represented by G-p^K, and 

x 

6 
its moment about the axis of the cylinder by G-p a ^K. So 

that the sum of the moments of all such forces of torsion in 
respect to the whole section aoibl will be represented by 

6 d 

(t -Zp a _iK, or by G-I, if I represent the moment of inertia 

x x 

of the section about the axis of the cylinder. Now these 
forces are in equilibrium with P; therefore, by the principle 
of the equality of moments, 

Ta=GI 7 (689). 

If r represent the radius of the cylinder, I=f-/' 4 (Art. 
85.). Substituting this value, representing by L the whole 
length of the cylinder, and by © the angle through which 
its extreme section CD is displaced or through which OP is 
made to revolve, called the angle of torsion, and solving in 
respect to ©, 

Thus, then, it appears that when the dimensions of the 
cylinder are given, the angle of torsion varies 
directly as the pressure P by which the torsion 
is produced; whence, also, it follows (Art. '.'7.' 
that if the cylinder, after having been deflected 
through any distance, be set free, it will oscil- 
late isochronously about is position of repose, 
the time T of each oscillation being represented 
in seconds (equation 76) by the formula 

(- *C \ 
^-^y I (®a) ; in which expression 




TORSION. 54S 

(®a) represents the length of the path described by the 
point P from its position of repose, so that the moving force 
upon the point P, when the pressure prdncing torsion is 
removed, varies as the path described by it from its position 
of repose. 

The above is manifestly the theory of Coulomb's Torsion 
Balance.* W represents in the formula the weight of the 
mass supposed to be carried round by the point P, and the 
inertia of the cylinder itself is neglected as exceedingly 
small when compared with the inertia of this weight. 

The torsion of rectangular prisms has been made the sub- 
ject of the profound investigations of MM. Cauchyf, Lame, 
et Clapeyron^:, and Poisson.§ It results from these investi- 
gations | that if b and c be taken to represent the sides of the 
rectangular section of the prism, and the same notation be 
adopted in other respects as before, then 

e = 3PLa(f + c') 

(orb C J 



M. Cauchy has shown the values of the constant G to 
be related to those of the modulus of elasticity E by the 
formula 

G=§E (693). 



In using the values of G deduced by this formula from 
the table of moduli of elasticity, all the dimensions must be 
taken in inches, and the weights in pounds. 



433. Elasticity of torsion in a solid having a circular 
section of variable dimensions. 

Let ah represent an element of the solid contained by 

* Illustrations of Mechanics, Art. 37. 

■| Exerciccs de Matin-mat iques, 4 e annee. 

t Crelle's Journal. § Mnnoires de l'Academie, tome viii. 

| Navier, Resume des Lecons, <tc, Art. 159. 



•50 



TORSION. 



planes, perpendicular to the axis, whose dis 
tance from one another is represented by 
the exceedingly small increment ±x of the 
distance x of the section ah from the fixed 
section AB, and let its radius be repre- 
sented by y\ and suppose the whole of the 
solid except this single element to become 
rigid, a supposition by which the conditions 
of the equilibrium of this particular element 
will remain unchanged, the pressure P re- 
maining the same, and being that which 
produces the torsion of this single element. 
Whence, representing by Ad the angle of 
torsion of this element, and considering it 

a cylinder whose length is ^a?, we have by equation (689), 

substituting for I its value fry% 

Passing to the limit, and integrating between the limits 
and L, observing that at the former limit 0=0, and at the 
latter &=&. 




2Pa rdx_ 
^G~J y K 



. (694.) 



If the sides AC and BD of the solid be straight lines, its 
form being that of a truncated cone, and if r r and 7\, repre- 
sent its diameters AB and CD respectively ; then 



Ako, 



dx 
dy 



T — T 

' i ' a 



u r 2 r, 



»•,— »v 



' 1 '2 



TORSION. 551 



434. The rupture of a cylinder by torsion. 

It is evident that rupture will first take place in respect 
to those elements of the cylinder which are nearest to its 
surface, the displacement of each section upon its subjacent 
section being greatest about those points which are nearest 
to its circumference. If, therefore, we represent by T the 
pressure per square inch which will cause rupture by the 
sliding of any section of the mass upon its contiguous sec- 
tion,* then will T represent the resistance of torsion per 
square inch of the section, at the distance r from the axis, at 
the instant when rupture is upon the point of taking place, 
the radius of the cylinder being represented by r. Whence 
it follows that the displacement, and therefore the resistance 
to torsion per square inch of the section, at any other dis- 
tance p from the axis, will be represented at that distance by 

-£, the resistance upon any element aK, by - pAK, and the 

sum of the moments about the axis, of the resistances of all 

T T 

such elements, by - 2p 2 AK, or by - I, *r substituting for I 

its value (equation 64) by -^TV. But these resistances are 
in equilibrium with the pressure P, which produces torsion, 
acting at the distance a from the axis ; 

:.Ya=iT*r* (696). 

It results from the researches of M. Cauchy, before referred 
to, that in the case of a rectangular section whose sides are 
represented by b and c, the conditions of rupture are deter- 
mined by the equation 

p ^ T (TO)i-'-- (697) - 

The length of a prism subjected to torsion does not affect 
the actual amount of the pressure required to produce rup- 
ture, but only the angle of torsion (equation 690) which 
precedes rupture, and therefore the space through which 

* Or the pressure per square inch necessary to shear it across (Art. 406.). 
f Navier, Itesume d'un (Jours, &c. Art. 107. 



TOE- 

the pressure most be made to act, and the amount ofvroKE. 

I ice rupture. 
ling to M. Cauehy. the modulus of rupture by tor- 
sion T is connected with 3 of rupture by trail; 
6train by the equation 

T=*S 




PAET VI. 
IMPACT* 



435. The impact of two bodies whose centres of gravtts 
move in the same right line, and whose point of con- 
tact is in that line. 

From the period when a body first receives the impact of 
another, until that period of the impact when both move for 
an instant with the same velocity, it is evident that the sur- 
faces must have been in a state of continually increasing 
compression : the instant when they acquire a common velo- 
city is, therefore, that of their greatest compression. When 
this common velocity is attained, their mutual pressures will 
have ceased if they be inelastic bodies, and they will move 
with a common motion ; if they be elastic, their surfaces 
will, in the act of recovering their forms, be mutually 
repelled, and the velocities will, after the impact, be dif- 
ferent from one another. 



436. A BODY WHOSE WEIGHT IS "W,, AND WHICH IS MOVING 
IN A HORIZONTAL DIRECTION WLTH A UNIFORM VELOCITY 
REPRESENTED BY Y„ IS IMPINGED UPON BY A SECOND BODY 
WHOSE WEIGHT IS "W* 2 , AND WHICH IS MOVING IN THE SAME 
STRAIGHT LINE WITH THE VELOCITY V 2 I IT IS REQUIRED TO 
DETERMINE THEIR COMMON VELOCITY Y AT THE CNSTANT OF 
GREATEST COMPRESSION. 

Let/j represent the decrement per second of the velocity 
of W, at any instant of the impact (Art. 94.), or rather the 
decrement per second which its velocity would experience 
if the retarding pressure were to remain constant ; then wili 

* Note (v), Ed. A PP . 



554 IMPACT. 

w 

— x -f 1 represent (Art. 95.) the effective force upon W, ; and if 
f\ be taken to represent, under the same circumstances, the 
increment of velocity received by W 3 , then will — -f % repre- 
sent the effective force upon AV 2 . Whence it follows, by the 
principle of D'Alembert (Art. 103.), that if these effective 
forces be conceived to be applied to the bodies in directions 
opposite to those in which the corresponding retardation 
and acceleration take place, they will be in equilibrium with 
the other forces applied to the bodies. But, by supposition, 
no other forces than these are applied to the bodies : these 
are therefore in equilibrium with one another, 

W W 

"Y A g f * (699) * 

Let now an exceedingly small increment of the time from 
the commencement of the impact be represented by a^ and 
let ^v l and &v 2 represent the decrement and increment of 
the velocities of the bodies respectively during that time, 

.-.(Art. 95.)f^t=A Vl ,f t st=Av % ; 

;. (equation 699) W t . a«-W, . ±v, ; 

and this equality obtaining throughout that period of the 
impact which precedes the period of greatest compression, it 
follows that when the bodies are moving in the same direc- 
tion 

W 1 (V-Y)=W 2 (Y-Y 2 ) (TOO); 

since Y,— Y represents the whole velocity lost by W, during 
that period, and Y— Y 2 the whole velocity gained by AV 2 . 

If the bodies be moving in opposite direccions, and their 
common motion at the instant of greatest compression be in 
the direction of the motion of W„ then is the velocity lost 
by AY X represented as before by (Y,— Y); but the sum of 
the decrements and increments of velocity communicated to 
W 2 , in order that its velocity Y 2 may in the first place be 
destroyed, and then the velocity Y communicated to it in an 
opposite direction, is represented by (V, + V). 

.\W 1 (V 1 -Y)=W,(V,-r--V). 

Solving these equations in respect to Y, we obtain 



IMPACT. 555 

WY + WY 

Y -^wr (701 > ; 

the sign ± being taken according as the motions of tht. 
bodies before impact are both in the same direction or iri 
opposite directions. 

If the second body was at rest before impact, Y 2 =0, and 

WY 

Y =wptr, cm* 

If the bodies be equal in weight, 

v=«v,±v s )- 

The demonstration of this proposition is wholly indepen- 
dent of any hypothesis as to the nature of the impinging 
bodies or their elastic properties ; the proposition is there- 
fore true of all bodies, whatever may be their degrees of 
hardness or their elasticity, provided only that at the 
instant of greatest compression every part of each body 
partakes in the common velocities of the bodies, there being 
no relative or vibratory motion of the parts of either body 
among themselves. 



437. To DETERMINE THE WORK EXPENDED UPON PRODUCING 
THE STATE OF THE GREATEST COMPRESSION OF THE SUR- 
FACES OF THE BODIES. 

The same notation being taken as before, the whole work 
accumulated in the bodies, before impact, is represented 

WW 

by \ — i Y x 2 -f i — - Y 2 2 ; and the work accumulated in them 

at the period of greatest compression, when they move with 

W a + W 

the common velocity Y, is represented by £ — a Y a . 

j 
Now the difference between the amounts of work accumu- 
lated in the bodies in these two states of their motion has 
been expended in producing their compression; if, there 
fore, the amount of work thus expended be represented b}" 
u, we have 

WWW +W 



556 IMPACT. 

or substituting for V its value from equation (701), and 
reducing, 

^(w^Wj^VJ . . . . (703). 

This expression represents the amount of work iiermanently 
k>:>t in the impact of two inelastic bodies, their common 
wl'-citY after impact being represented by equation (701). 
If \V 3 be exceedingly great as compared with W l5 

•"JXV.TVJP .... (704). 



438. TWO ELASTIC BOOTES IMPINGE UPON ONE ANOTHER I IT IS 
REQUIRED TO DETERMINE THE VELOCITY AFTER IMPACT. 

If the impinging bodies be perfectly elastic, it is evident 
that after the period of their greatest compression is passed, 
they will, in the act of expanding their surfaces, exert 
mutual pressures upon one another, which are, in corres- 
ponding positions of the surfaces, precisely the same with 
those which they sustained whilst in the act of compression ; 
whence it follows that the decrements of velocity expe- 
rienced by that body whose motion is retarded by this 
expansion of the surfaces, and the increments acquired by 
that whose velocity is accelerated, will be equal to those 
before received in passing through corresponding positions, 
and therefore the whole decrements and increments thus 
received during the whole expansion equal to those received 
during the whole compression. 

Now the velocity lost by AV\ during the compression is 
represented by (V l — V); that lost by it during the expan- 
sion, or from the period of greatest compression to that 
when the bodies separate from one another, is therefore 
represented by the same quantity. But at the instant of 
greatest compression both bodies bad the velocity V ; the 
velocity -i\ of W 1 at the instant of separation is therefore 
V— (Yj — V), or 2V— V,. In like manner, the velocity 
gained by W, during compression, and therefore during 
expansion, being represented by (Y^Y,), and its velocity 
at the instant ot greatest compression by V, its velocity v t 
at the instant of separation is represented by V + ^T^Oj 
or by 2V=fV 2 , the sign =f being taken according as the 



IMPACT. 557 

motion of the bodies before impact was in the same or 
opposite directions. 

Substituting for Y its value in these expressions (equation 
701), and reducing, we obtain 

If the bodies be perfectly elastic and equal in weight, 
0,=Y 2 , v i =Y 1 ; they therefore, in this case, interchange their 
velocities by impact ; and if either was at rest before impact, 
the other will be at rest after impact. 

If W 2 be exceedingly great as compared with W l5 v t = 
— Y 1 ±2Y 2 , %=±Y 3 . In this case v x is negative, or the 
motion of the lesser body alters its direction after impact, 
when their motions before impact were in opposite direc- 
tions ; or when they were in the same direction, provided 
that 2Y 2 be not greater than Y r 



439. If the elasticities of the balls be imperfect, the force 
with which they tend to separate at any given point of the 
expansion is different from that at the corresponding point 
of the compression ; the decrements and increments of the 
velocities, produced during given corresponding periods of 
the compression and expansion, are therefore different; 
whence it follows that the whole amounts of velocity, lost 
by the one and gained by the other during the two periods, 
are different : let them bear to one another the ratio of 1 to e. 
Now the velocity lost during compression by W, is under 
all circumstances represented by (Y x — Y); that lost during 
expansion is therefore represented, in this case, by e (Y, — Y); 
therefore, v l = V—e(V 1 —V)=(l + e)V—eY l . In like man- 
ner, the velocity gained by W„ during compression is in all 
cases represented by (Y=F Y„) ; that gained during expansion 
is therefore represented by e(Y=f V 2 ) ; therefore, « 9 =Y+ 
<<V=F V,)—(l + e)V^eV,. Substituting for Y, and reducing, 

y- O^-^V.iq + ^W.V, (707V 

V^ + W, V ; ' 

_±( W Q -,AY 1 )Y 2 + (1 + ^)W 1 Y 1 (708) 
W, I -\V a 



558 IMPACT. 



440. In the impact of two elastic bodies, to determine 
the accumulated work, or one half the vis viva, lost 
by the one and gained by the other. 

The vis viva lost by W, during the impact is evidently 

represented by -^Y *-- ^ a = JlifV'-v') = -^ J Y a - 
1 J 9 9 9 K J 9 X ' 

1(1 + e)Y- eY, \> 1 =^l j(l - e*)Y x * + 2e(\ +e)YY- (l + ef 

Y 2 } =^(l+e)(Y-Y){Y 1 {l-e) + Y(l + e)\. 

Substituting in this expression its value for Y (equation 
701) reducing and representing by u x one half the vis viva 
lost by W, in its impact, or the amount by which its accumu- 
lated work is diminished by the impact (Art. 67.), 

(l+e)W,V,j ....(709). 

Similarly, if w a be taken to represent one half the vis viva 
gained by W 2 , or the amount by which its accumulated work 
is increased by the impact, then 

(l+e)W l Y l \ (710). 

441. Let u be taken to represent the whole amount of the 
work accumulated in the two bodies before their impact, 
which is lost during their impact. This amount of work is 
evidently equal to the difference between that gained by the 
one body and lost by the other; so that u=u x — u v Substi- 
tuting the values of u x and u 2 from the preceding equations; 
and reducing, we obtain 

(W)w,av,(v,tv,)- nm 

This expression is equal to one half the vis viva lost during 
the impact of the bodies. If the bodies be perfectly elastic, 
€=1, and u=0. In this case there is no real loss of vis viva 



IMPACT. 559 

in the impact, all that which the one body yields, during the 
impact, being taken up by the other.* 

442. In the preceding propositions it has been &upposed 
that the motions of the impinging body, and the body im- 
pinged upon, are opposed by no resistance whatever during 
the period of the impact. There is no practical case in 
which this condition obtains accurately. If, nevertheless, 
the resistance opposed to the motion of each body be small, 
as compared with the pressure exerted by each upon the 
other, at any period of the impact, then it is evident that all 
the circumstances of the impact as it proceeds, and the mo- 
tion of each body at the instant when it ceases, will be very 
nearly the same as though no resistance were opposed to the 
motion of either, f 

443. As an illustration of the principle established in the 
last article, let it be required to determine the space through 

* It has been customary, nevertheless, to speak of a loss of vis viva in the 
impact of perfectly elastic bodies. This loss is in all such cases to be under- 
stood only as a loss experienced by one of the bodies, and not as an absolute 
loss. When the impinging bodies are perfectly elastic, it is evident that the 
one flies away with all the vis viva which is lost in the impact by the other. 

f Let Pi and P 2 represent resistances opposed to the motions of two im- 

Wx W 2 

pinging bodies whose weights are Wx and W 2 ; also let — /x, and — / 2 re- 
present the effective forces upon the two bodies at any period of the impact ; 
then, by D'Alembert's principle, 

Wy^.w, 0; 

9 9 

or representing by t the time occupied in the impact, up to the period of 
greatest compression, by V their common velocity at that period, and by v t 
and v 2 their velocities at any period of the impact, and substituting for /i and 
/ 2 their values (equation 72), 

Wx dv x _, W 2 dvt 

— - — ±i— -=- — r^—v. 

g at 9 at 

Transposing and integrating between the limits and t, 

t 

9 9 J 

Now if Px and P 2 be not exceedingly great, the integral in the second member 
of the equation is exceedingly small as compared with its other terms, and may 
be neglected ; the above equation will then heroine identical with equation 
(700). 



560 imp 

h a nail will be driven by the blow of a hammer; and 
supposed that the resistance opposed to the driving 
of the nail is partly a constant resistance overcome at its 
point, and partly a resistance opposed by the friction of the 
mass into which it ie driven upon it arying in amount 

directly with the length of it a?, at anytime imbedded in the 
wood. Let this resistance be represented by a+fix\ then 
will the work which must be expended in driving it to a 
depth D be represented (Art. 51.) by 

D 

/7a + oVK/.r, or by (aD + ^D*). 

I) 

Let W, represent the weight of the nail, and Ythe velocity 
with which a hammer whose weight is W J must impinge 
upon it to drive it to this depth, and let the surfaces of the 
nail and hammer both be supposed inelastic; then will the 
work accumulated in the hammer before impact be repre- 

W 

sented by J — ^ 2 , and the amount of this work lost during 

the impact by the compression of the surfaces of contact will 

1 / "W AY \ 
be represented (equation 711) by — L ~ - v V\ The work 

remaining, and effective to drive the nail, is therefore repre- 
sented by the difference of these quantities; and this work 
being that actually expended in driving the nail, we have 

1 VW, 1 



= 2aD-f-oD a .... (712; ; 

y " it- " i 

solution of whi 
mined. 



by the solution of which quadratic equation, D may be deter- 



444. Two SOLID PRISMS HAVE A COMMON AXIS ; THE EXTREM- 
ITY OF UIEM BE8TS AGAINST A FIXED SURFACE, AND 
\ TREMITY RECEIVES THE IMPACT, IX A HORI- 
BTTAL DIBEI [TON, OF THE OTHER PRISM: IT IS REQUIRED 
TO DETERMINE THE COMPRESSION OF EACH PRISM, THE LIMITS 
Of PERFl ' ELASTICITY NOT BEING PASSED IN THE IMPACT. 

Let \Y represent the weight of the impinging prism, and 



IMPACT. 561 

Y its velocity before impact ; L x and L 2 the lengths of the 
prisms before compression ; E x and E 2 their moduli of elas- 
ticity ; K x and K 2 their sections ; l x and Z 2 the greatest com- 
pressions produced in them respectively by the impact ; 
then will the amounts of work which must have been done 
upon the prisms to produce these compressions be repre- 
sented (equation (486) by the formulse 

i-j- - and I 2 L 2 % 

and the whole work thus expended by 
'KEi 2 KJEX 






W 

But this work has been done by the work -§• — Y 2 , accumu- 
lated (Art. 66) before impact in the impinging body, and 
that work has been exhausted in doing it ; 

,, ( w + w )=1 w v . 

Moreover, the mutual pressures upon the surfaces of con- 
tact are at every period of the impact equal, and at the 
instant of greatest compression they are represented respec- 
tively (equation 485) by j ' 1 and t 2 2 ; 

. K 1 E 1 Z 1 _ K 2 E 2 Z 2 __p (113) 

L t L 2 

Eliminating Z 2 between this equation and the preceding, and 
reducing, 



- V 1U.E, 



+ A) ff ; 



in which expressions l x represents the greatest compression 

36 



502 IMPACT. 

of the prism whose section is K,, and P the driving pressure 
at the instant of greatest compression. 

445. The mutual pressures P of the surfaces of contact at 
any period of the impact. 

Let I represent the space described by that extremity of 
the impinging prism, by which it does not impinge : it is 
evident that this space is made up of the two corresponding 
compressions of the surfaces of impact of the prisms ; so that 
if these be represented by Z, and Z 2 , then l=d x +l % . But 

(equation 713) ^j^ 1 , ^=0 9 ' therefore l = T ' (i£E" 1 " r 

M. 

•••«(&+«) (716> 

446. A measure of the compressibility of the prisms. 

If X be taken to represent the space through which that 
extremity of the impinging prism by which it does not 
impinge will have moved when the mutual pressure of the 
surfaces of contact is 1 lb. ; or, in other words^if X repre- 
sent the aggregate space through which the prisms would 
be compressed 'by a pressure of 1 lb. ; then, by the preced- 
ing equation, 

X maybe taken as a measure of the aggregate compressi- 
bility of the prisms, being the space through which their 
opposiU extremities would he made to approach one another 
by a pressure of 1 lb. applied in the direction of their 
length. , , 

If \ and \ represent the spaces through which the 
prisms would severally be compressed by pressures of 1 lb. 

applied to each, then \=jn|r s ^KY, ' therefore X==X > + 

\, or the aggregate compressibility of the two prisms is 
equal to the sum'of their separate compressibilities. 



IMPACT. 563 



447. The work u expended upon the compression of the 
prisms at any period of the impact. 



The work expended upon the compression l x is repre- 
ss 
sented by i -y-- l l* ; or substituting its value for l x (equation 

713), it is represented by ij^~-Y 2 . And, similarly, the work 

expended on the compression \ is represented by ixr^ P 2 > 

H 2 .b a 

therefore ^=Jl^r-^ +tf4t IP*; or substituting for P its 

value from equation (716), 



448. The velocity of the impinging body at any period of the 
impact, the impact being supposed to take place vertically. 

It is evident that at any period of the impact, when the 
velocity of the impinging body is represented by v, there 
will have been expended, upon the compression of the two 
bodies, an amount of work which is represented by the 
work accumulated in the impinging body before impact, 
increased by the work done upon it by gravity during the 
impact, and diminished by that which still remains accu- 

W W 

mulated in it, or by £— V*+WJ— £— v\ 

if if 

Representing, therefore, by u the work expended upon 

W 

the compression of the bodies, we have -J — Y a + *W7— ■ 

if 

J— v=u. 

g 

Substituting, therefore, for u its value from equation 
(718), 

"W "W / T T \ — 1 



• 



:>:: .-. t. 



k+AT'.-C' 



19). 



ulue in tern- 



. It is 




The Pile Deivee. 

evident that the pile will not begin to be 
driven undl a period of the impact is at- 
tained, when the pressure of the ram npon 
gether with the weight of the 
pile, Lb the resistance opposed fa 

motion by the coherence and the fricti- i 
the mass into which it is driven. Let this 
resistance be represented by P ; let V repre- 
sent the ve. :he ram at the instant of 
impact, and r its velocity at the instant when 
the pile begins to move, and "vV.. AV 

-he ram and pile ; then, since the 
pile will have been at rest during the whole 
: he intervening period of the impact, since 
rr. the mntual pi the sur- 

faces of are at the instant of motion 

red bv P— TT.. we 1- quation 

•- 



■-- 



p_W 



KA 






w. 



i p-w, ;. . . . - 



If the value of v determined by this equation be not a 
possible quantity, no motion can be communicated to the 
pile by the impact of the ram; the following inequal: 
the: -ndition necessary to the driving of the pi 



**>'&+!§ 



'-- 



er the pile has moved through any given distance, one 

>n of the work accumulated in the ram before its 

impact will have been expended in overcoming, through 

that distance, the resistance opposed to the motion of the 



IMPACT. 565 

pile; another portion will have been expended upon the 
compression of the surfaces of the ram and pile ; and the 
remainder will be accumulated in the moving masses of the 
ram and pile. The motion of the pile cannot cease until 
after the period of the greatest compression of the ram and 
pile is attained ; since the reaction of the surface of the pile 
upon the ram, and therefore the driving pressure upon the 
pile, increases continually with the compression. If the 
surfaces be inelastic, having no tendency to recover the 
forms they may have received at the instant of greatest 
compression, they will move on afterwards with a common 
velocity, and come to rest together ; so that the whole work 
expended prejudicially during the impact will be that 
expended upon the compression of the inelastic surfaces of 
the ram and pile. If, however, both surfaces be elastic, 
that of the ram will return from its position of greatest 
compression, and the ram will thus acquire a velocity rela- 
tively to the pile, in a direction opposite to the motion of 
the pile. Until it has thus reached the position in respect to 
the pile in which it first began to drive it, their mutual 
reaction Q will exceed the resistance P, and the pile will 
continue to be driven. After the ram has, in its return, 
passed this point, the pile will still continue to be driven 
through a certain space, by the work which has been accu- 
mulating in it during the period in which Q has been in 
excess of P. When the motion of the pile ceases, the ram 
on its return will thus have passed the point at which it 
first began to drive the pile : if it has not also then passed 
the point at which its weight is just balanced by the elas 
ticity of the surfaces, it will have been continually acquiring 
velocity relatively to the pile from the period of greatest 
compression; it will thus have a certain velocity, and a 
certain amount of work will be accumulated in it when the 
motion of the pile ceases: this amount of work, together 
with that which must have been done to produce that com- 
pression which the surfaces of contact retain at that instant, 
will in no respect have contributed to the driving of the 
pile, and will have been expended uselessly. If the rani in 
its return has, at the instant when the motion of the pile 
ceases, passed the point at which its weight would jn^i be 
balanced by the elasticity of the surfaces of contact, its 
velocity relatively to the pile will be in the act of diminish- 
ing; or it may, lor an instant, cease at tlu- instant when the 
pile ceases to 'move. In this Last case, the pile and ram, tor 
an instant, coming to rest together, the whole work accurou 



566 IMPACT. 

I in the impinging ram will have been usefully expended 
in driving the pil< _ uly thai :h the remain- 

f the guri - Inced : which 

3 lees than that due to the weight of the ram. 
Thi>. ther ad se in which a maxi- 

mum useful effect is produced by the ram. The following 
article contains an analytical die - >n of these conditions 
under their most general form. 



450. Apru by anotl I thi 

U 

wired r 

' ' ' ' 

'her rejects tte 
Artidi 44^. 

Let f. and f % represent the additional velocities which 
would be lost and acquired per second (see Art. 
by the impinging prism and the prism 
impinged upon, if the pressures, at any instant 
operating upon them, were to remain from that 

TV , W 
instant constant ; then will —f.. —{f, reptw 

the effective forces upon the two bodies (Art 1 
or the pressures which would, by the principl 
D'Alembert, be in equilibrium with the unbal- 
anced pressures upon them, if applied in opposite 



directions. 

X w the unbalanced pressure upon the system 
sed of the two prisms is represented bv 

w-vr.-p. 

:^f- yy,=W t +W,-P (723); 

also the unbalanced pressure upon the prism PQ=W,+ 
q — P, where Q represents the mutual pressure of the prisms 

; .^y-W,+Q-P (75 

Let A have been the position of the extremity B of the 

impinging prism at the instant of impact : and let a?, repre- 

the space through which the __ gate length BP of 

the two prisms has been diminished since that period of the 



IMPACT. 567 

impact, and x % the space through which the point P has 
moved ; then (equation 716) 

M& + i&r=x (725) - 

f](rp I rp \ 

Also AB=^+ai, ; therefore velocity of point B= — b; - ^ 

(Art. 96); therefore /*=5 + 5=g ;+ /, 

Substituting these values of f x and Q in equations (723) and 
(724), and eliminating/!, between the resulting equations, 



tfx. 

dt 



--i(w^w) x ^% w 



r= 



Integrating this equation by the known rules,t we obtain 

x,=A sin. yt+B cos. rtf + Hrrfr (727) ; 

y vv 3 

in which expression the value of y is determined by the 
equation 

and A and B are certain constants to be determined by the 
conditions of the question. Substituting in equation (724) 
the value of Q from equation (725), and solving in respect 
to/,, 

/.-rfc^+'f 1 ^)* w- 

Substituting for x x its value from equation (727), and for f t 

d 2 x 
its value -^, and reducing, 

d% Ag . ^Bg I P \ 

__ = _J sin . r , + _ cos. 7 *+ \l-t(rqPW;)^ 

Integrating between the limits and t, and observing that 

ci r p 
when £=0, -7-7= ; the time being supposed to commence 
dt 

with the motion of the prism PQ, 

» Art. 96. Equations (72) and (74). 
\ Church's Int. Cal. Art. 183. 



563 IMI'ACT. 

Integrating a second time between the same limits, 

* 3 = ^vH^ ^ - sin - ^) + w;x7 & — cos - rQ + 
p 



*(i-wrFr>----w>- 



JNow when the motion of the second prism ceases -7^=0 ; 
whence, if the corresponding value of t be represented by T, 

A(l-cos. 7 T) + B sin. y T + ( 1 - w ^ y ) W a x 7 T^ 0.(731). 

To determine the constants A and B, let it be observed 
that the motion of the prism QP cannot commence until the 
pressure Q of the impinging prism upon it, added to its own 
weight W a , is equal to the resistance P opposed to its motion. 
So that if c be taken to represent the value of x x (i. e. the 
aggregate compression of the two prisms) at that instant, 

then, substituting for Q its value from equation (725), - + 

Now since the time t is supposed to commence at the 
instant when this compression is attained, and the prism PQ 
is upon the point of moving, substituting the above value of 
c for x l in equation (727), and observing that when x=c. 

t=0, we obtain (P— TV 2 )X=BH — ^-; whence by substitu 

tion from equation (728), and reduction, 

(P-W,-W,)y / P i 

So long as the extremity P, of the prism impinged upon, 
is at rest, the whole motion of the point B arises from the 

dx 
compression of the two prisms, and is represented by — jj- t 

cCt 



IMPACT. 569 

fjsy> 

The value of — -, when £=0, is represented therefore by a 
ctt 

(equation 721). Differentiating, therefore, equatioD (727), 
assuming £=0, and substituting v for-^-j we obtain ^=/A; 

whence it appears that the value of A is determined by 
dividing the square root of the second number of equation 
(721) by /• 

Substituting for A and B their values in equations (731-3) 

^( 1 -cos. r T) + X Wj (_Z_ : -l) s in. r T + 

Reducing, and dividing by the common factor of the two 
last terms, 

Substituting for A and B their values in equation (730), and 
representing by D the value of a? 2 , when t=T, 

.... (735). 

The value of T determined by equation (734) being sub- 
stituted in equation (735), an expression is obtained for the 
whole space through which the second prism is driven by 
the impact of the first.* 

* The method of the above investigation is, from equation (7'2t'>\ nearly the 
same with that given by Dr. Whewell, in the last edition of his Mechanics; the 
principle of the investigation appears to be due to Mr. Airey. l\' the value 
of y, as determined by equation (728), were not exceedingly great, then, since 
the value of T is in all practical cases exceedingly small, the value of yT would 
in all cases be exceedingly small, and we might approximate to the value of 
T in equation (735), by substituting for cos. yT and sin. yT, the twc first termi 
of the expansions of those functions, in terms of yT. 



EDITORIAL APPENDIX. 



Note (a). 

Besides its direction defined (Art. 1), we have also to take 
into consideration, in estimating the effects of a force, its 
'point of application^ or the point of the body where it acts, 
either directly or through the medium of some other body, 
as a rigid bar, or an inextensible cord in its line of direction ; 
the point on its line of direction towards which the point of 
application has a tendency to move ; and finally the inten- 
sity, or magnitude of the force as expressed in terms of some 
settled unit of measure. 

Note (b). 

This result of experiment also admits of the following 
proof: Let A be the point of appli- 

< — «? *» — ** — «ec;p cation of a force P, and let this point 

p/ be invariably connected with another 

point B, in the line of direction towards which A tends to 
move from the action of P ; suppose now two other forces 
P, and P 2 , each equal to P, to be applied ; the one at A, in 
a direction opposite to P, and the other at B, in the same 
direction as P ; the introduction of these two equal forces, 
acting m opposite directions,will evidently in nowise change 
the direction or intensity of P ; but as P, is equal and oppo- 
site to P its effect will be to balance the action of P at A, 
whilst it leaves P a to exert an action at B precisely the Bame 
as P was exerting at A before the introduction of P, and P a . 



Note (<?). 
Suppose two forces P t and P 9 applied to the same point A, 

HI 




572 rdtx 

the direction of the one being AB, that of the 
other AC; do was these forces make an angle 
with each other, it is evident, as the point 
application can move but in one direction, and 
is 6olici I to move - I) and C at 

the same time, that it must move in Borne 
direction which i* coincident with neither of 
these'; this direction, it i.- equally evident, 
mnst he in the same plan. lirectione AB and A( 

there is no argument in favor of a direction assumed ex 
to the plane and on one side of it which would not equally 
apply to a symmetrical direction assumed on ti. 
it is also evident that this direction must be Borne one AF 
within the angle formed by AB and A( . for the point, if 
solicited by P, alone, would take the direction AB, and 
cannot take a direction to the left of BI). a? there i- no force 
that solicits it on that side, and, fur like reasons, cannot take 
one to the right of CE, it must therefore take the one 
ssifl led somewhere within the angle BAC. 
Now suppose further that P, and P, are equal, it is evi- 
dent that the direction assigned to their resultant, or that of 
the motion of their point of application, must be the one 
which bisects the angle BAC, for the argument fur any 
direction on the left of this line would be equally cogent for 
the like position on the other side. 

If Pj and P, are unequal then will the direction of their 
resultant divide the angle BAC unequally, the 
smaller portion being next to the greater force : 
for suppose Pj divided into two portions, one 
nt' which P shall be equal to P. : : P and P, can 
be replaced by their resultant R,, the direction 
«>f which AF bisects the angle F>A( ' : we shall 
then have two forces R, and the remaining 
portion of P l5 the resultant of which R must lie 
somewhere within the angle BAF, aud there- 
fore nearer to I J X than to P„ : but I£ is the resultant of the 
two forces P, and P„. Tl . &c. 

Hence it is seen that tw a whose directions form an 

angle between them and m . . - . have a resultant ; 2nd, that 
the direction of this resultant lies in the plane of the two 
forces : 3d, that it passes through the point where the direc- 
tions meet, and lies within the angle contained between 
them: 4th, that it bisects this angle when the forces are 
equal; 5th. that when the forces are unequal it divides this 
angle unequally, the smaller portion being next to the greater 
force. 





EDITORIAL APPEJTDIX. 0<G 

Now as the two forces P a and P 2 can be replaced by their 
resultant R, and as the effect of this will be the 
same if applied at any point F in its line of 
direction as at the point of application of the 
two forces, it is evident, if we transfer P x and P 9 
also to F, preserving their new parallel to theii 
original directions, that they, in turn, can be made 
to replace P. It thus appears that the point of 
application of two forces may be transferred to 
any point of the line of direction of their result- 
ant without changing the effects of these forces, provided 
their new directions are kept parallel to their original ones. 
It is upon the preceding propositions, in themselves self- 
evident, that the mode of demonstration, known as Duchay- 
la's, of the proposition, termed the parallelogram of forces, 
or of pressures, is based. 

Note (d). 

When two parallel forces are applied to two points inva- 
riably connected, their resultant can be found by applying 
the propositions in (Arts. 1, 2, 3). 

Let P x and P 2 be two parallel forces applied at the points 
A and B invariably connected, as by a 
rigid bar. Let two equal forces Q 1 and 
Q 2 be so applied, the one at A the other 
at B, as to act in opposite directions 
fryQ.t along AB. These two will evidently 
have no effect to change the action of 
P x and P 2 . Now the two forces P, and 
Q l applied at A will have a resultant R„ 
the intensity and direction of which can 
be found by the preceding method. In like manner the 
resultant R 2 of P 2 and Q 2 can be obtained. Now the forces 
being replaced by their resultants, the equilibrium will still 
subsist, and the effect will remain the same whether K, and 
K 2 act at A and B, or at o their point of meeting. Bu1 as 
R, and R 2 can each be replaced by their components at any 
point of their direction, let these components be transferred 
to the point o. In this position Q x and Q a will destroy each 
other, whilst P, and P 2 will act in the same direction alone 
<?Cand parallel to their original ones, with an intensity equal 
to their sum P,-j-P 2 . 

Now from the similar triangles \<>(\ rom ; and Bo< , son, 
there obtains, 




' ~ 4- EDITORIAL APPEMDDL, 

om : r | : : oC : CA. 

1 : . I 

Multiplying the r roportions, there obtains, 

P, : P, : : CB : CA. 
and 

P : : P, : P t +P, :: CB : CA : GB+< B. 

From this we see that two parallel forces acting in the 
same direction, 1st. have a resultant which is equal to their 
sum ; 2nd, that the direction of this resultant is parallel to 
that of th a; 3d, that it divides the line joining the 

points of application of the - into parts reciprocally 

rtional to th - ; ±th, that either I - to the 

resultant as u Q of the line between the resultant and 

-her force is to the total distance between the 
appl: "th, that the foregoing prop -old true 

ion of the line AB with respect to the 
parallel forces and their resultant. 

When the two forces act in opposite directions at the 

points A and B. by following 

rain the 

two resultants E, and 
^-'" which being prolonged to 

' ¥/ their point of meeting o we 

•q, can again replace them by 

their components P.. Q.. and 
P.. Q : : which P, and P : . 
acting parallel to their ori- 
ginal positions but in opposite dnv ill have for their 
resultant Y x — P.. 

N w prolonging the direction of this resultant until it 
t8 Aj5 prolonged at C. there obtains as in the preceding 
ease, from the similar triang'. rwn, and ~BoC, son? 

om : mr :: oC : CA. or P. : Q. :: C : CA. 
m: on :: CB : -C. orQ, : P s :: CB : 
hence, 

P t : P, : P.-P, :: CB : CA : CB-CA or AB. 



EDITORIAL APPENDIX. 



57! 




Remark. — Although it may be assumed, as self-evident; 
that any resultant can be replaced 
by two equivalent components, 
without disturbing the equilibrium, 
and that each of these in turn 
may be replaced by two other 
equivalent components, and so on 
for any number of components ; still like compositions and 
resolutions of forces are of such frequent occurrence in esti- 
mating the pressures, or strains on the various points of any 
mechanical contrivance, as a machine, a frame of timber, 
&c, arising from a resultant pressure, that the student can- 
not be too familiar with the processes of effecting such com- 
positions and resolutions. 

To show by a simple illustration this truth, let the result- 
ant AR be replaced by its two equivalent components AP : 
and AP 2 in any assumed positions ; and let each of these 
components be resolved into two others, AQ 1? AR X for AP a ; 
and AQ 2 , AR 2 for AP 2 , taken respectively perpendicular and 
parallel to AR. Now it is evident, from the figure, that the 
two components AQ l5 AQ 2 of this last resolution are equal 
and opposite in direction, and therefore destroy each other ; 
whilst the two AR l5 AR 2 act in the direction of AR, and 
their sum is equal to AR. The same may in like manner be 
shown for any number of sets of components by which AR 
might be replaced. 



Note (e). 



If the point o from which perpendiculars are drawn to the 
directions of two forces P, and P 2 , is 
taken on the direction of their re- 
sultant, then will m t F 1 =m i F^. 

For from o draw the perpendicu- 
lars om, on, to P, and P 2 , and join 
the points m and n of their inter- 
section. The quadrilateral Aw»on, 
having the angles at m and // righ.1 
angles, can be inscribed in a circle, therefore the two angles 
at m and A, subtended by the chord on, will be equal. In 
the triangles mon and ABC, the angle o is the supplement 
of the angle A of the quadrilateral, and B, being tne adja- 
cent angle of the parallelogram, is also the supplement of A ; 




576 EDITORIAL APPENDIX. 

the two triangles, having two angles equal, are similar, 
therefore, 

All: BC : : om : on, or P, : P, : : om : on ; 

hence 

P, x om=Y^ x on. Therefore, &c. 

From this proposition the relations of two parallel forces 
to their resultant can be readily deduced from the limiting 
case of the angle mon of the triangle ; for from the two simi- 
lar triangles there obtains as before 

P 2 : Pj : P or AC : : om : on : mn. 

Now as this is true for any value of the angle 0, when it 
becomes 1S0°, the forces P„ P 3 having the same direction, 
and their resultant P become parallel ; the perpendiculars 
om and on become portions of the line mn ; and, as ?nn=om 
+ 0?i, it follows, from the above proportion, that P=P 1 + P 2 . 

"When P, and P 2 have opposite directions, we can suppose 
the force P„ for example, and its perpendicular turned about 
the point in the plane of the forces until the point m falls 
on the prolongation of on on the opposite side from 0, in 
which position P x and P 2 again become parallel, but act in 
opposite directions. During this rotation of P 1? the resultant 
still passes through 0, and there still obtains 

P 2 : P, : : P : om : on : mn ; 

but, as mn now is equal to om—on, it follows, from the 
proportion, that P=P 2 — P I# Hence the same relations 
between P„ P 2 and P as already established, Kote (d). 



KOTE {/). 

Otherwise, since in any number of forces in equilibrium 
either of them is equal and directly opposed to the resultant 
of all the rest, the whole may be replaced by these two 
without disturbing the equilibrium. If now through the 
point of application of these two we draw any two lines at 
right angles to each other, and then resolve each of the two 
forces into two components parallel to these two lines, it 
will be at once seen, from the diagram, that the like com- 



EDITORIAL APPENDIX. 577 

ponente will be equal and opposite to each other, and this 
would evidently be the same tor the components of the ori- 
ginal forces resolved in the same manner, otherwise there 
would be a resultant for all the forces, which is contrary to 
the supposition of an equilibrium. 

Ei "iiuwh..- —As this method of resolving a system of forces 
into sets of components parallel to any assumed rectangular 
axes, in order to determine their algebraical values, is of* 
frequent use, in simplifying the numerical calculations 
necessary in the applications of the principles of statics, the 
student should make himself perfectly familiar with the pro- 
positions that precede and follow Art. 11. 



Note (g). 

Otherwise, join DE which will be parallel to AC, thus 
forming with it and the lines AD and CE two cqui-angular 
triangles, from which there obtains 

DE: DG :: AC: AG; 

but DE=£AC, therefore DG=£AG=-£DA. 

Koto (A). 

Otherwise, join Gil which, as AG and Oil intersect, will 
be in the same plane with them and with the line AC. As 
AM and CG are respectively J of the lines drawn from A 
and C to the middle of I>1>, it follows that Gil is parallel to 
AC and forms with it and the lines AG and CH, by th<' : 
intersection at K, the two equi-angular triangles G 12 ^ 1 
and AKC, from which there obtains 

Gil :GK:: AC: AK, 
but GH=iAC, therefore GK=iAK=iAG. 

Note (*). 

As the methods employed in (Art. \->. &c.) to represent 
by geometrical diagrams, what are termed the laws of 
motion, or the relations which exist at any two given 

87 



:' 



57S EDITOBIAL APPENDIX. 

instants between the velocity, the space, and the time of a 
body's motion, although very simple in themselves, are 
sometimes found to offer difficulties to the Btudent, particu- 
larly as to the representation of spaces by areas, a few addi- 
tional marks on this point may nor be here misplac 

Taking, in the first place, the case of a body M moving in 
^ v a rectilinear path from A towards 

B with a uniform motion. Ac- 
cording to the definition, the 
body will move over the suc- 

i ! ■ ; £ cessive equal portions Ah, he, ed, 

&c, of its path in equal sin 
sive portions of time, however small or great these portions 
may be. Taking now any portion of time as a unit, as 
a minute^ &c, and supposing Ah the portion of its path, or 
the space through which the body has moved during this 
unit, Ah will represent what is termed the velocity, or rate 
of motion of the body ; and when the path itself is expn 
in terms of any linear unit, as a foot, a yard, a mile, &c, the 
number of these units in Ah will measure the velocity ; for 
example, if the unit of path, or space is a foot, and there 
were four of these units in Ah. and the unit of time 
second, then the velocity would be termed a velocity of four 
feet per second, &c. Supposing the body to start from A, 
with this velocity, it will successively move over distances, 
each of four feet in length, along its path, in sncc< •- 
seconds of time ; consequently any distance, or space, as Ad, 
will be equal to Ah taken as many times as the number of 
seconds elapsed from the time the body started from A until 
it reached d\ or, in other words, the number of units in the 
space Ad is expressed by the abstract number obtained by 
multiplying the number of units in the velocity by the num- 
ber of units in the time. This, like all other similar pro- 
ducts, can be expressed algebraically, or geometrically ; but 
by whatever symbol expressed, the signification is the same. 
For example, on any two lines, as AB and AC, taken at 
right angles, set off any number of equal parts as Aft, 5c, 
cL &c, as units of time, and on AC any number also of 
equal parts, which may be the same in length, or otherwise, 
as those on AB, to represent the units in which the velocity 
is expressed. Suppose the latter to be composed of the four 
units A//>, //>/'. &c. : and that the number of units of time 
considered is three : on the lines A //, AC construct the 
rectangle AD ; then is the area of the rectangle said to 
express the space corresponding to the velocity and time 






EDITORIAL APPENDIX. 579 

here assumed ; that is, the number of units in area of this 
rectangle, expressed in terms of the unit of area on Ab and 
Am for example, is equal to the number of units of space. 
In like manner the area of the rectangle AE expresses the 
space corresponding to the velocity and the time Ae, &c. 
In uniformly varied motion, as the velocity increases ii: 

the same proportion as the time 
-x* 5 " increases, or, in other words, as 
jt^\ the augmentations of the velocity 

for equal intervals of time is the 

same, these relations between the 
I B times, velocities and spaces, can, 

in like manner, be expressed by 
a geometrical diagram as follows : On any line, as AB, set 
off' a number of equal parts as Ab, be, cd, &c, to represent 
equal intervals of time ; at the points b, c, d, &c, having 
drawn perpendiculars to AB, set off on them distances bm, 
en, do, &c, to represent the corresponding velocities; in 
which cn=2bm do=3fo?i ; or Ad : Ae : Ab :: do : en : bm, 
&c. Now, as the same relations obtain between all the dis- 
tances set off on AB and their corresponding perpendicu- 
lars, it follows that the line AC, drawn through the points 
m, n, o, &c, is a right line, and that the triangles Abm, Acn, 
&c, are therefore similar. As the relations between the 
times and velocities are true, however great, or however 
small the equal portions of time may be assumed, let us sup- 
pose these portions, as Ab, be, cd, to be taken so small that 
the velocity of the body during any one of them may be 
considered uniform, and as a mean between what it actually 
is at the commencement and end of this portion ; that is en 
and do, for example, representing the actual velocities at the 
beginning and end of the interval of time represented by cd, 
then -J- {cn + do) represents the mean, or uniform velocity 
during this interval. This being premised, the number of 
units of space over which the body will pass whilst moving 
with a uniform velocity, expressed by £ (en + do), during the 
interval cd, will be represented, according to the preceding 
proposition, by cdx% (cn + do), but this also expresses the 
area of the trapezoid cdno ; and as the same is true for all 
the like trapezoids it will also be true for their sums, or for 
the triangles, as Ado and Afq for example, the areas of which 
are equal to the sum of the areas of the trapezoids of which 
they are composed. Supposing the body to move from a 
state of rest with a uniformly accelerated motion, and that 
at the intervals of time, represented by Ad ami A/', its 



580 EDITORIAL AITENDIX. 

respective velocities are do andjfo, then will the number of 

units of space which the body will have moved over in these 
two intervale be respectively expressed by the number of 
units of area in the triangles Ado and Afq. As the trian- 
gles are similar their areas are as the squares of their like 
Bides; it therefore follows that in uniformly varied motion, 
the spaces are as the squares of the times, or as the squares 
of the velocities. 

As do represents the velocity acquired during the time 
A//, supposing the body to have moved from a state of rest, 
and the number of units of area in the triangle Ado repre- 
sents the corresponding number of units of space, it follows, 
that if the body had moved, during the same interval, with 
the velocity do which it actually acquired in it, the number 
of units of space it would then have passed over would have 
been represented by the number of units of area in the rect- 
angle A<9, constructed on Ad and do. But, as the area of 
the rectangle is double that of the triangle, the space that 
would have been passed over in the supposed case would 
have been double that passed over in the actual case. 

If we take any portion, as Ae, to represent the unit of 
time, then the corresponding perpendicular ep w T ill represent 
the velocity, or the quantity f used in (Arts. 46. 47) fol- 
lowing. 

Note (J). 

As the propositions under this head, and those under the 
heads of Accumulation of Work in a Moving Body (Art. 64) 
and Principle of Vis Viva (Art. 129) constitute the basis of 
what may be termed Industrial Mechanics, or the applica- 
tions of the principles of abstract mechanics to the calcula- 
tion of the effects of motive power transmitted by machines 
and employed to produce some useful median ieal end, it is 
very important that the student should have a clear and 
definite apprehension of their signification in this point of 
view. Work, as here defined, supposes two conditions as 
essential to its production : a continued resistance, or obstacle 
removed by the action of a force, and a motion of the point 
of application of the force in a direction opposite to that in 
which the resistance acts. Its measure is expressed by the 
product arising from multiplying the number of units of the 
resistance, or of its equivalent force directly opposed to it, 
by the number of units of path which the point of applica- 
tion of this force has described during the interval consi- 



EDITORIAL APPENDIX. 581 

dered, in which the force acts to overcome the resistance. 
It follows that the work will be when this product is ; 
that is, when either of the factors, the resistance, or its equi- 
valent force, or the path described, is 0. 

In estimating work, that which is external and which alone 
generally we have the means of measuring, is alone consi- 
dered. For example, if with a flexible bar a person attempts 
to push before him any obstacle, the first effect observed will 
be a certain deflection of the bar, during which the hand, at 
one end of the bar, will have moved forward a certain dis- 
tance in the direction of the point of application at the other, 
producing an amount of work which is expressed by the 
product of the pressure, or force exerted by the hand, sup- 
posing this pressure to remain constant during this period, 
and the path described, in the line of direction of this pres- 
sure, by the point where it is applied. During this period, 
as the obstacle to be moved has remained at rest, no path 
has been described by the point where the bar rests against 
it, therefore, according to our definition, no work has been 
done upon the resistance. The effect produced by the 
pressure has been simply to bend the bar, and the work is 
therefore due only to the resistance offered by the molecular 
forces of the material composing the bar to the force that 
tends to bend it. This portion of the work, although in this 
case we have the means of measuring it, being what may be 
termed internal, is not taken into the account in estimating 
that due to the resistance to be overcome, which would have 
been the same had a perfectly rigid bar been used instead of 
the flexible one. 

In like manner, when an animal carries a burthen on his 
back from one point to another on a horizontal plane no 
work is produced according to our definition ; for no resist- 
ance has been overcome in the direction in which the bur- 
then has been carried, and therefore the product that repre- 
sents the work is 0. The work in this case, as in that of the 
flexible bar, is internal; and similar to that arising from a 
burthen borne by an animal whilst standing still ; and there- 
fore although both of them may be very useful operations 
and have a marketable value, still they can neither be mea- 
sured by the standard by which it is agreed to estimate 
work. 

Every mechanical operation performed by machinery pre- 
sents a case of work. Take for example the simple opera- 
tion of planing, in which the hand moves a plane, which is 
but a rigid bar to which is fixed an iron tool like a chisel for 



EDITOBIAL APPENDIX. 

removing from the edge, or Burfacc 

of a board. In this case the resistance offered, and which is 
ibly in tin- Bame direc ie the power applied, is that 
arising from the cohesion of the fibres of the material, 
is measured by the pressure applied ; the path which the 
]x>int of application of the iron tool describes is the 
that described by the hand: and the work will be expr 
by the product of these two elements, each estimated in 
terms of its own unit of measure. The case of the common 
grindstone presents an example of a rather more complicated 
character. Here the instrument to be ground is pn 
against the periphery of the stone with sufficient fore.- t-» 
a certain resistance to any power however applied to 
put the si"one in motion. The direction however in which 
this resistance acts at the point of application is in the 
direction of the tangent to the periphery at this point, and, 
in one revolution of the stone, it will describe a path equal 
in length to the circle described by the point of application. 
The work therefore for each revolution will be the product 
of the resistance, estimated in the direction of the tangent, 
and the circumference described by the point of application. 
It may be as well to remark, in this place, that although 
the work done in overcoming the molecular re-istances of 
the materials by means of which the action of a force or 
-ire is transmitted, as in the example above cited of a 
flexible bar. i^ not taken into account in estimating the 
external work, there are cases in which this work constitutes 
the entire work done, and which again is reproduced in 
external work : as for example in the cases of the common 
bow used for projecting arrows, and the springs by which 
the machinery of ><>me time-pieces is moved. In each of 
these the resistance offered by the molecular forces of the 
material is overcome by the action of some external force, 
whose point of application is made to describe a given path ; 
by tlii- action a certain amount of work is expended in 
bringing the spring to a certain degree of ten-ion which, 
when the force is withdrawn, will reproduce the same amount 
of external work in an opposite direction to that in which the 
force acted. 

XOTE (k). 

The work of a pressure of constant intensity acting in the 

same direction a- the path described by its point of applica- 

may be represented by a geometrical diagram in the 



EDITORIAL APPENDIX. 



5S3 



same way as that used for representing the space described 
by a body moving with a uniform velocity in any given 
time ; by constructing a rectangle, one side of which repre- 
sents the number of units of force, the other the number of 
units of path ; the number of units of area of the rectangle 
will express the number of units of work. 



Eote (I). 

The method given (Art. 51) for estimating, by a geometri- 
cal diagram, the work of a pressure which varies in inten- 
sity at different points of the path described in its line of 
direction by its point of application, finds its application and 
has to be used whenever there is no geometrical law of con- 
tinuity by which the pressure can be expressed in terms of 
the path ; and, even when such a law obtains, it is some- 
times found to be a more convenient method of obtaining an 
approximate value of the amount of work than the more 
rigorous one expressed by the formula 



A. 



■'J '///////I 'D 



S, 

in which TJ can be rigorously found whenever P, which being 
a function of S can be expressed algebraically in terms 
of it. 

As an example of these two methods of estimating the 
work of a variable pressure, acting in the 
direction of the rectilinear path described by 
its point of application, let the familiar case 
of the action of steam on the piston of the 
steam-engine be taken. 

Let ABCD represent the steam-tight cy- 
linder in which the piston is driven from the 
position at a, at one end, to c at the other, in 
the direction of the axis ac, of the cylinder, 
by means of the pressure of the steam on the 
end of the piston. Let us suppose thai the 
steam acts with a constant pressure, repre- 
sented by P t , whilst thepistOD is driven through the portion 
ha of the path, and, having reached tin's point, tin- commu- 
nication between the cylinder and the boiler being then cut 



is 



584: EDITORIAL APPENDIX. 

off, that the steam already admitted acts, through the re- 
mainder of the path described by the piston, by what is 
termed its expansive force, In which the pressure continually 
decreases as the piston approaches the point c. Let us sup- 
that the law of variation of this pressure on the piston 
at different points is such that the pressures at any two 
points are inversely proportional to their distances from the 
point ". P, then denoting the pressure when the piston is 
at p, let P denote the pressure when it has reached another 
point o at a distance S from 0, and S, and S, denote the 
lengths ac and 00, then according to the above law there 
obtains 

P, : P :: S : S„ therefore Y=yJL 

o 

Let the elementary portion of the path be denoted by rZS, 
then by multiplying the variable force by the elementary 
path there obtains 

Pd9=P a fsdS, 

which may be termed the elementary work, or in other 
words, the work done whilst the variable pressure acts 
through the elementary path, during which period the vari- 
able pressure may be regarded as constant. 

To obtain the total work whilst the variable pressure acts 
from b to c, or through the path S 2 — S„ there obtains 

Si s 2 

U =f?dS = ?&f ^ = P 1 S 1 (l0g.e 8,-log.e S,). 

s, s, 

If instead of the exact work due to the expansive force of 
the steam, and which is given by the foregoing formula, an 
approximate value only was required, it could be obtained by 
a geometrical diagram as follows. 
Tv Having set off to any scale a num- 
ber of units representing the path 
/"•. calculate the pressures at the 
-...._. points b, e, and at the middle point 
0, for a first approximation. That 
\ e at b will be simply P, ; that at 0, 

P,|, and that atJ.P,-^^- 



EDITORIAL APPENDIX. 585 

Having drawn perpendiculars to be at b, 0, and c, set off on 
them the distances bm, on and cj? respectively equal to the 
corresponding pressures, estimated in terms of the unit of 
pressure, and according to any given scale. Join mn and 
np / the number of units of area, in the figure thus formed, 
estimated in units of path and pressure, will be an approxi- 
mate value of the required number of units of work. 

The greater the number of parts into which be is divided and 
the corresponding pressures calculated, the nearer will the 
enclosed area approach to the true value of the work. 

The mean pressure, or that force which, acting with a 
constant intensity along the same path as that described by 
the point of application of the variable pressure, would give 
the same work, is found either by dividing the result of the 
integration by S a — S 13 or by dividing the area in the last 
method by be. 



Note (m). 

As an example of the manner of obtaining the work done 
by a constant pressure acting always 
in parallel directions whilst its inclina- 
tion to the path described by its point 
of application is continually varying, 
let the well known mechanism of the 
crank arm and connecting rod be taken. 
Let O be the centre around which the 
crank arm is made to revolve, by the 
application of a constant pressure P„ 
transmitted through a connecting rod 
CD, all of whose positions during the motion are parallel to 
the diameter AB. The path described by the point of ap- 
plication C will be the circumference of which OC is the 
radius, and the inclination of P : to this path will be the 
variable angle DCN, between its direction and the tangent 
to the circle at C, of which the variable angle AOC, thai 
measures the inclination of the crank arm to the diameter 
AB, is the complement. Denote this last angle by a, and 
the length of the crank arm OC by b: Now decomposing 
P, into components in the direction of the tangent ( \ and 
the radius OC, we obtain for the first P, sin. o, and for the 
second P, cos. a, of which P, sin. a is alone effective to pro- 
duce work, since P, cos. a acts constantly towards the fixed 
point without describing any path in the direction o( its 



tj: 


*'*», 


b C 


_|T, 






06-^ 


\ m - 


i-yt 


1 




Py 



5SG EDITOBIAL APPENDIX. 

►n. But the elementary path described by the point 
of application is evidently baa, t lie infinitely small arc On of 
circle. The elementary work of the variable component 
-in. a will therefore be expressed by 

Pj sin. a x Ida. 

Hie total work for any portion of the }:>ath, as AC, will 
therefore be 

r a . 

I PjSin. abda=Ffi(l — cos. o)=P l Jver. sin. a. 
o 

and for a=r, it becomes 

P^^orP.xAB; 

a result which might have been foreseen, since AB is the 
path described by the point of application of P, in its line 
of direction, whilst the actual path is the Mini-circumfe- 
rence ACB. 

ks On=bda i if through n a perpendicnlar nm ie drawn to 
C\), the line of direction of P 1? the distance Cm is evidently 
the projection of the elementary path actually described on 
the line of direction of P„ and is therefore the corresponding 
elementary path of P x in its line of direction ; but Cm=Cn 
sin. a=bda sin. a. Denoting AB by h, then Cm=dh; and 
there obtains 



and 



d/i=bda sin. a ; and P, dh=T > 1 Ida sin. a; 
f V/Jh = TJi = f Vfi sin. a da=V, x 25. 



A result the same as is shown to obtain by the preceding 
proposition. 

To find the mean, or constant pressure which, acting in 
the direction of the circular path, would produce the same 
amount of work as the variable force does in acting through 
the semi-circumference ; call Q this mean force, its path 
being tt5, its work will be Q x~b ; and as this is to be equal 
to the work of P, sin. a, there obtains 

Q x tt^P, x 25, hence Q=P, - =06366 P, nearly, 
for the value of the force. 



EDITORIAL APPENDIX. 587 

It may be well to observe here that the mean pressures 
have no farther relations to the actual pressures than as 
numerical results which are frequently used instead of the 
actual pressures to facilitate calculations ; and also as a 
means of comparing results, or work actually obtained from 
a force of variable intensity, at different epochs of its action, 
with what would have been yielded at the same epochs by 
the equivalent mean force. 

To show the manner of making the comparison in this 
case, let us take the two expressions for the quantity of work 
due the mean force, and also to the variable component, for 
a portion of the path corresponding to any angle a. Since 

o 
Q=P 1 -, its work corresponding to a will be 
it 

The corresponding work of the variable component P, sin. a 
will be 

P^l — cos. a). 

The difference therefore between these two amounts of work 
will be 

P^-P^l-COS. a)=-?J> |^-l + cos. a\ 
Now this difference will be for the following values of a, 

a=0, a— -, and a—ix. 
2i 

The maximum value of this difference can be found by the 
usual method of differentiation and placing the first differ- 
ential coefficient equal to 0. Performing this operation, 
there obtains 

2 
sin. a~ - =0*6366; 

it 

the corresponding values of a being respectively 
a = 0-21964: 7r,and a = n — 0-21964 tt. 

Substituting these values of a and the corresponding values 
of cos. a in the preceding expression for the difference there 
obtains, for the first, 



588 EDITOR] LL APPENDIX. 



\\b (??-l+eo8. a\ = T t l [2x0-21964-1+ |/l- 1) = 

-f 0-21039 IV>; 
and for the second, 



P5 (^-1+cos. a )=P^/2-2x 0-21964:- l-^/l- i) = 

- 0-21039 PA 

From these two expressions i1 is seen that the greatest excess 
of the work of the mean force over that of the other would 
be 4- 0-21039 P,5= +0-1052 xP^or about T \ of the total 
work of P, corresponding to the path 25; whilst that of the 
work of Pi over the meanforce, represented by — 0-2lu39P,6, 
is the Bame in amount. 

It' now we Buppose the direction of the constant force P, 
to be changed, when its point of application reaches the point 
13, so as to act parallel to the direction BA until the point of 
application arrives at A, it is clear that the work of P, due 
to the path described from B to A will also be expressed by 
P,x_V;, m> that the work due to an entire revolution of the 
point of application will be P, x4//. As the mean force will 
evidently be the same for the entire revolution of the point 
of application, it follows that the greatest positive, or nega- 
tive excess, as stated above, will be - 052b' x P,4J, or ^ of 
the work for one entire revolution. 

It is thus seen that although the work of the effective 
variable componenl P, sin. a of P, is not, like that of the mean 
force, uniform for equal paths, still it at no time falls short 
of nor exceeds "the work of the meanforce by more than 
about 2V of the entire work for each revolution. Were any 
mechanism, as that for jumping water for example, so 
arranged that either the constant force P,, or a mean force 
equal to 0*6366Pj, acting as above described, were applied 
to it, the quantity of water delivered by the one would at no 
time exceed, in any one revolution, that delivered by the 
other by more than ^ of the total quantity delivered by 
either during the entire revolution. 



Note (n). 

If P„ \\<v example, were the resultant of the other pres- 
sures, its component P a cos. o a would be equal to the al^e- 



EDITORIAL APPENDIX. 588 

braic sum of the components P x cos. a l5 P 3 cos. a 3 , &c, of the 
other pressures P 1? P 3 , &c. ; the work therefore of P 2 , esti- 
mated in the direction of the given path AB, and corres- 
ponding to any portion of this path, will be equal to the 
algebraic sum of the work of the other pressures P„ P 3 , &c, 
corresponding to the same portion of the given path. 



Note (o). 

Since at the point E, taken as the point of application, the 
line of direction of the pressure becomes a tangent to the 
arc described with the radius OE, it follows that the infi- 
nitely small arc described with the radius OE may be taken 
for the infinitely small path described by the point of appli- 
cation in the direction of the tangent. Denoting by da the 
infinitely small angle described by the radius OE, then 
OE x da will express the infinitely small path, or arc ; and 
P x OE^Za will represent the elementary work of the 
pressure. 

If the pressure remains constant in intensity and direction 
during an entire revolution of the body about 0, then will 
the work of P for this revolution be represented by 
P x circum. OE. 



Note (j>). 

The term living force is more generally used with us by 
writers on mechanics instead of its Latin equivalent vis < 
to designate the numerical result arising from multiplying 
the quantity denominated the mas3 of a body by the square 
of the velocity with which the body is moving at any 
instant. It will be readily seen that this product does not 
represent a pressure, or force, but the numerical equivalent 
of the product of a certain number of units of pressure and a 
certain number of units of path. The one magnitude being 
of as totally a distinct order from the other as an in- 
different from a lino, and therefore having no common unit 
of measure. 

Besides this expression, which serves uo other really ose- 
fnl purposes than a- a name to designate a certain numerical 
magnitude which is of constant occurrence in the Bubjecl of 
mechanics, there is another also of frequent use, termed 



EDITORIAL APPENDIX. 

- wtion, wnich ie the'product of the maps and 

tlu* velocity j or v. This ie also termed the dynamical 

measuri oi a force in contradi6tmctioD to pressure, as usually 
I, which is termed the statical measure of a force. 



Xote (q). 

In estimating the accumulated work in the pieces of a 
machine which have either a continuous or a reciprocating 
motion of rotation it is necessary to find expressions for the 
moments of inertia of these pieces with respect to their axis 
of rotation, and this may, in all cases, be done, within a cer- 
tain degree of approximation to the true value, by calculat- 
ing separately the moment of inertia of each of the compo- 
nent parts of each piece and taking their sum for its total 
moment of inertia, on the principle that these moments may 
be added to or subtracted from each other in a manner 
similar to that in which volumes, or areas are found from 
their component parts. 

In making these approximate calculations, which in many 

s are intricate and tedious, it will be well to keep in view 

the two or three leading points following, with the examples 

given in illustration oi some of the more usual forms of 

rotating pieces. 

1st. The general form for the moment of inertia of a body 
rotating around an axis parallel to the one passing through 
its centre of gravity as given in equation 58, (Art. 79) is 

Now if the distances of the extreme elements of the body 
from the axis passing through its centre of gravity are small 
compared with that of /(. the distance between the two axes, 
the second term I of the second member of this equation 
may he neglected with respect to the first, and /rM be taken 
as the approximate value of the required moment. This 
consideration will find its application in many of the cases 
referred to. as. for example, in that of finding the moment of 
inertia of the portion of a solid, like the exterior flanch of the 
beam of a steam-engine, the volume of which maybe approx- 
imately obtained by the method of Guidinus (Art. 39.). In 
this case, A representing the area of the cross section of the 



EDITOPwIAL APPENDIX. 591 

flanch, and s the path which its centre of gravity would 
describe in moving parallel to itself in the direction of the 
flanch around the beam, any elementary volume of the 
flanch between two parallel planes of section will be ex- 
pressed by Ads. Now the moment of inertia of this elemen- 
tary volume from equation 58 is 



■.Aff*da+I; 



in which the first term of the second member, which 
expresses the sum of the elementary volumes Ads into the 
squares of their respective distances r from the axis of rota- 
tion, may be taken as the approximate value required ; inas- 
much as I, the sum of their moments of inertia with respect 
to the parallel axes through their centres of gravity, may be 
neglected with respect to the first term. The problem will 
therefore reduce to finding the moment of inertia of the line 
represented by s, which would be described by the centre of 
gravity of A, with respect to the assumed axis of rotation, 
and then multiplying the result by A. 

2nd. As the line s is generally contained in a plane per- 
pendicular to the axis of rotation, and is given in kind, as 
well as in position with respect to this axis, being also gene- 
rally symmetrically placed with respect to it, its required 
moment of inertia may, in most cases, be most readily 
obtained by finding the moment of inertia of s separately, 
with respect to two rectangular axes contained in its plane, 
and taken through the point in which the given axis of rotation 
pierces this plane, and then adding these two moments. 

The moment of inertia of a line taken in this way with 
respect to a point in its plane has been called by some 
writers the polar moment of inertia. 

This method is also equally applicable to finding the mo- 
ment of inertia of a plane thin disk revolving around an axia 
perpendicular to its plane, and to solids which can be divided 
into equal laminseby planes passed perpendicular to the axis 
of rotation. 



(re 1 ) The moment of inertia of the arc of a parabola with 
respect to an axis perj>< -ndicuhtr to fh< plane of the cwrve at 
a given point on the axis of the curve. 

Let BAC be the given arc ; A the vertex of the parabola ■ 



m 



EDITORI A L API' E M 1 1 X . 




H R the point on its axis at which the 
axis is taken. Through R draw 



the chord PQ. Represent the 

i> chord EC of the given arc by b] 
its corresponding abscissa AD by a; 
and AB by c. Let y represent the 
ordinate pq^ and x the correspond- 
ing abscissa of any element dz of the arc. 

From the preceding remarks, the moment of inertia of dz 
with respect to the axis AD will be expressed by ifdz ; and 
that of the entire arc BAC by 

AB \b 

2 fed** = | fe(? + teayfdy ; 



as from the equation of the parabola, y*= 
By integration f 






ib 





in which Z is the length of the arc BAC. 

In like manner the moment of inertia of dz with respect 
to the chord PQ is 

/ (c — xfdz 
and for the entire arc BAC, 

AB ib 



which integrated as above, 

Ia = I 1 + "32^" + 32 . 64 <?cN &Z ~~~ 

V" a + 32a 9 V 64 

* Church's Int. Cal. Art. 199. \ Ibid. Art. 160. 



EDITORIAL APPENDIX. 593 

From the preceding remarks, the moment of inertia of Z, 
with respect to the axis at the point K perpendicular to tho 
plane of Z, is 



i,+i. 



- \ + 32«c 32. Ua'o'l 



a3 ( 1 + i6^) 2 (i -&?-*)■ 

The value of Z in the above expressions is 



Z= i {V + ie<)*+ £ log. e (y + I \/v + 16a' ) * 

Each of the preceding expressions may be simplified, and 
an approximate value obtained, sufficiently near for practical 
applications, when the ratios of b and c to a are given. For 
example, when b / \a there obtains 

Z=2« + ^log. £ y; 



the terms omitted, being small fractions with respect to 
unity, do not materially affect the result. 

Having found the moment of inertia of a parabolic curve, 
that of a parabolic ring of uniform cross section, taken per- 
pendicular to the direction of the 'curve at any point, and 
having its centre of gravity at its point of intersection with 
the curve, can be obtained by simply multiplying I x + I a by 
8 the area of the given cross section. 



(ft 1 ) The moment of inertia of the segment of a parabola with 
respect to an axis perpendicular to its plane at a <jir< n 
point of the axis of the curve. 

Let BAC be the given segment ; A the vertex ; AD tho 

* Church's Int. Cal. Art. 199. 

38 



594: 



EDITORIAL APPENDIX. 




an elementary 
A(2y)'(fo. That 



axis of the curve; and D the point on 
the axis with respect to which the mo- 
ments of inertia arc estimated. Denote 
the chord BC by b ; the abscissa AD 
by a. 

By (Art. 81.) the moment of inertia of 
area pq with respect to AD is j2{pq) s dx= 
of the segment therefore will be 



v> 



\ = &ffyfa=<hf^tfdy 



A «*"• 



In like manner the moment of inertia of an elementary 
area asps, with respect to the axis BC, is -J- (ps) s dy=i(a—xy 
dij. That of the segment therefore will be 



+6 v> 

\ = if (a-xyd*, = i f(a- ^yjdy 



tYt^- 



oF 



JLA- 
l o 5 



a'b. 



.-.1, + L. 

From this last expression we readily obtain the moment 
of inertia of a disk having the segment for it s base and its 
thickness represented by c, with respect to an axis at D per- 
pendicular to its base by simply multiplying I, + 1 2 by c ; or 

&+ 1 8 ) c = A <d?c + ffoa'bc = 1 abc (*?+« **); 
in which f abc = V, the volume of the disk. 



(c 1 ) The moment of inertia of a parabolic disk, or prism, 
with respect to an axis parallel to the chords which termi- 
nate the upper and lower bases and midway between the 
chords. • 

Let pq be an elementary volume of the disk contained 
between two planes parallel to the base 
BC of the disk. Adopting the same 
notation as in the preceding article, the 
volume pq is expressed by 




2y 



dx. 



The moment of inertia of this elemen- 
tary volume with respect to an axis 
through its centre of gravity and parallel 
toBCVis (Art. 83) 



EDITORIAL APPENDIX. 595 



V . 2y .0 .dx \<? + {dxf 



1 2 



and the moment of inertia of the same volume with respect 
to the axis, parallel to the one through its centre of gravity, 
taken on the base BC of the disk and midway between the 
upper and lower chords is (Art. 79 Eq. 58) 



2y .c .dx {<? + (dx) % \ + 2y . c . dx (a-xf ; 



the moment of inertia of the entire disk with respect to the 
same axis is 

a a 

;. I = T u f 2y .c.dx \c u + (dx)*] + f 2y . c . dx (a-x)\ 



Substituting for x and dx in terms of y, omitting the term 
containing (dx)*, and integrating as indicated, there obtains, 

in which Y=%abc. 



(d') The moments of inertia of a right prism with a trape- 
zoidal base with respect to axes perpendicular and parallel 
to the base at the middle point of the face terminated by the 
broader side of the trapezoid. 

Let AGHC be the trapezoid forming the base of the prism. 
Represent the altitude EF of the trape- 
zoid by a ; AG by b ; CH by b 1 ; and the 
height GB of the prism by c. Let jp^ De an 
elementary volume of the prism between 
two planes parallel to the face AB and 
^ at a distance Ee=x from the face CD. 
JB From C drawing Cc parallel to II G there 
obtains 

pr= w .Ac=-(b-b); 

x 
\ ps=pr + rs = - (b — b 1 ) -f b\ 




EDITORIAL APPENDIX. 

The elementary volume j>q is therefore 
I x 



S 



The moment of inertia of pq with respect to an axis through 
ntre of gravity and perpendicular to the base of the 
prism is (Art. B 



and that of the entire prism with respect to an axis at F, the 
middle point of AG. and parallel to the preceding axis, is 



a 

omitting the term containing (dx)\ and integrating, as indi- 
cated, there obtains 

■i ! 24 b-b l V 

in which V=a J ~ J c. 



By a lib Derations the moment of inertia of the 

entire prism, with respect to an axis perpendicular to the 
preceding one at its middle point between the upper and 
lower bases of the prism, will be 






EDITORIAL APPENDIX. 



597 



(e ) The moment ■ of inertia of a right prismoid with rectan- 
gular bases with respect to an axis XY through the centre 
of gravity of the lower base and parallel to one of its 



Let AB = b, BC = c be the sides of the rectangle of the 
lower base ; ab = b\ be = & the sides 
of the upper base. Let pgrs be any 
section of the prismoid parallel to 
the lower base and at a distance x 
from it ; and let a be the altitude of 
the prismoid, or the distance between 
its upper and lower bases. 

From the relations between the 

^y dimensions of the prismoid there ob- 
tains (Art. d') 

a 

x /7 71N 71 xCb — b^ + ab 1 




qr=-(c-c)+c = 



x (c — &) + a& 



a 



and to express the elementary solid contained between two 
planes parallel to the base of the prismoid and at the height 
x above it, 



x (b — b 1 ) + ab 1 x(c — c 1 ) + ac 1 
a a 



dx. 



The moment of inertia of this solid, with respect to an 
axis xy through its centre of gravity and parallel to XY, ig 
(Art. 83) 

x(b — b l ) + ab 1 x(c — c l ) + ac 1 
x — ■ . dx 



T2 



"It 



x (c — &) + ac 



,1V2 

) + (dx) 



The moment of inertia of the prismoid (Art. 79 Eq. 58) is 



I : ITORIAL APPENDIX. 



r x(b — V)-\-ab l x(c — c*) + ac l 

I x ct x »r . 

J a a 

o 

omitting the term containing (da?)' and integrating as incb 

rated, there obtains, 

I = ±a {be + ib l &) + hatPc + be 1 ) + zhab(c l + 2c 1 c + 3c V + 4c*) + 

a ! 5 ab l (c + 2cV + 3c c*+ 4c 1 ). 

By integrating the expression for the elementary volume 
between the same limits, there obtains to express the volume 

< >t* the prismoid 

which is the formula usually given in mensuration. 

In each of the preceding examples, the quantities I, I,, &c. 
are expressed only in terms of certain linear dimensions ; to 
obtain therefore the moments of inertia proper these results 

most be multiplied by the quantity -, or the unit of mass 

corresponding to the unit of volume, in which p represents 
the weight of the unit of volume of the material and 
g = 32£ feet. 

Each of the above values of I may be placed under more 
simple forms for the greater readiness of numerical calcula- 
tion by throwing out such terms as will visibly affect the 
result in only a slight degree. But as such omissions depend 
upon the numerical relations of the linear dimensions of the 
parts no rule for making them can be laid down which will 
be applicable to all cases. 



(/*') The moment of inertia of a trip hammer. 

These hammers consist of a head of iron of which A repre- 
sents a side and A' a 
A ' a front elevation ; of a 

wk — [ El j |j)| fn nandle °f w °° d B > 

iTJ] Tct-'-p £->- ^ which is either of the 

AMi lU shape of a rectangulai 

parallelopiped, or of 



EDITORIAL APPENDIX. 599 

two rectangular prisraoids, having a common base at the 
axis of rotation C where the trunnions, upon which the 
hammer revolves, are connected firmly witli the handle by 
an iron collar. Another iron collar is placed at the end of 
the handle, and is acted on by that piece of the mechanism 
which causes the hammer to rotate. 

To obtain the moment of inertia of the whole, that of each 
part with respect to the axis is separately estimated and 
the sum then taken. 

The head A, A' may be regarded as a parallelopiped of 
which the side A', reduced to its equivalent rectangle by 
drawing two lines parallel to the vertical line that bisects 
the figure, is the end, and the breadth of the side A is the 
length. If then from the moment of inertia of this parallel- 
opiped that of the void #, or eye of the hammer, which is 
also a parallelopiped, be taiten, the difference will be the 
moment of inertia of the solid portion of the head. The 
moments of inertia of these parallelepipeds may be calcu- 
lated, with respect to the axis C, by first estimating them 
with respect to the axes through their respective centres of 
gravities G and g, parallel to 0, by (Art. 83) and then witli 
respect to C by (Art. 79. Eq. 58). Or if the moments of 
inertia with respect to G and g are small with respect to the 
product of their volumes and the squares of the distances 
GO and gC, then the difference of the latter products may 
be taken as the approximate value. 

The moment of inertia of the handle, if also a parallelo- 
piped, will be found with respect to C by (Arts. 79, 83). If 
it is composed of two rectangular prismoids, then the mo- 
ment of the parts on each side of the axis must be found by 
(«') and their sum taken. 

The moment of inertia of the trunnions and the iron hoop 
to which they are attached may be found by (Arts. 85, 87) 
and their sum taken. But as this quantity will be generally 
small with respect to the others it may be omitted. 

That of the hoop at the end of the handle may hv taken 
approximately as equal to the product of its volume and the 
square of the distance between the axis through its centre 
of gravity and that of rotation. 



(g') The moment of inertia of a cast Iron wheel. 
These wheels usually consist of an exterior rim A A' of 






EDITORIAL APPENDIX. 




uniform cross section con- 
nected with the boss, or nave 
C, ( ". which is a hollow 
cylinder, by radial pieces, or 
arms B, B', the cross section 
of which is in the form of a 
cross. Each arm having the 
same breadths at top and 
bottom in the direction of 
the axis of the wheel as those of the rim and nave which it 
connects ; the thickness perpendicular to the axis being 
uniform. The projection or ribs on the side of each arm, 
and which give the cross form to the section, being of uni- 
form breadth and thickness ; or else of uniform thickness 
but tapering in breadth from the nave to the rim. These 
ribs join another of the same thickness that projects from the 
inner surface of the rim. 

Represent by E the mean radius of the rim, estimated from 
the axis to the centre of gravity of its cross section; Jits 
breadth, and d its mean thickness ; Fits volume, and I its 
moment of inertia with respect to the axis; p the weight of 
its unit of volume, and g=3'2} feet ; then by (Art. 86) 



r=2~:TM. and I: 



FIT, 



omitting \cP as but a small fractional part of K\ 

Representing by l x the breadth of the arm at the axis, 
supposing it prolonged to this line ; b„ its breadth at the rim, 
supposing it prolonged also to the mean circle of the rim, d l 
its thickness ; Y x its volume ; 1\ its moment of inertia, then 



r x =Kd x 



\+\ 



and I, 



"*i& + «fl 



Representing by a x the breadth at bottom, a Q the breadth at 
top of the ribs, or projections on the sides of each arm, esti- 
mated also at the axis and mean circle of the rim ; d, their 
thickness ; F, their volume ; I 2 their moment of inertia : 
then by («r 

g ( 0! + ^ ±C / 



--R.7 a x + a * 



V n =Vul 



The sum 1 4- 1, -j- I a will be the moment of inertia of the 



EDITORIAL APPENDIX. 601 

entire wheel approximately, since the moment of inertia of 
the portions of the boss between the arms is omitted, this 
being compensated for by supposing the arms prolonged to 
the axis and to the mean circle of the rim. As the quanti- 
ties V^ F" 2 , Ij and I 2 are taken but for one arm, they must 
be multiplied by the number of arms to have the entire 
moment. 



{h') The moment of inertia of a cast iron steam engine beam. 

These beams usually consist of two equal arms symmetri- 
cal with respect to a 

^^___ ^#- [f ^ — _^ ^^ line a a through the 

b^^^^i&Mn^=^ ^. — , e^Sg jv f./ axis of rotation o. 

t ^ ===z == =~^ :i ^ - ====^ Each arm, a V a! and 

a o a\ consists of a 
parabolic disk of uniform thickness ; b and b' being the ver- 
tices of the exterior bounding curves, a a' their common 
chord, and ob, 6b' their axes. The disk is terminated on the 
exterior by a nanch B of uniform breadth and thickness. A 
rib C, either of uniform breadth and thickness, or else of 
uniform thickness, and tapering in breadth from the centre o 
to the ends b, b\ projects from each face of the plane disk 
along the axis b b'. The beam is perforated at the centre, 
near the two extremities and at intermediate points, to 
receive the short shafts, or centres around which rotation 
takes place. Around each of these perforations, projections, 
or bosses D', D", &c, are cast, co add strength and give a 
more secure fastening for the shafts. 

The beam being symmetrical with respect to a a\ it will 
be only necessary to calculate the moments of inertia of the 
component parts of each arm with respect to the axis, o and 
take double their sum for the total moment of inertia of the 
beam. These component parts are — 1st, the parabolic nanch; 
2nd, the parabolic disk of uniform thickness enclosed by the 
flanch; 3d, the rib on each side of the disk, running along 
the central line bb' ; 4th, the projections, or bosses D' &c, 
around the centres. 

The moment of inertia of the flanch will be calculated by 
(«') as its thickness is small compared with the other linear 
dimensions. That of the disk will be calculated by (ft'). 
That of the rib by (a'). Those <>t* the projections may be 
obtained within a suiheient degree of approximation by 



EDITORIAL APPENDIX. 

taking the product of their volumes and the squares of tlieii 
respective distances from the axis o. 

The sum of these quantities being taken it must be multi- 
plied by - as in the preceding cases ; m. being the weight of 
the unit of volume of the material. 



'Note (*). 

The increase of tension due to rigiditv and which is ex- 

D+E P 

pressed by * — " may be placed under the following 

form, 

c m . a + c m . & . P a c m (a + b . P a ) 

K R 

by writing c m . a for D, and c m . b for E, in which c repre- 
sents the circumference of the rope, and m the power to 
which c is raised. 

The increase of tension of any other rope whose circumfer- 
ence is c x bent over the same pulley and subjected to the 
6am e tension P a is, in like manner, expressed by 



R 

Now representing by T and T a the two values above for the 
respective increase of tension for c and c x there obtains, by 
dividing the one by the other, 



(?)"=■$■ ^ce T, = (5)"T; 



which expresses the rule given above for using the tables in 
calculating the increase of rigidity due to a cord whose cir- 
cumference is different from those in the tables. 



Note (f). 

As one of the chief ends of every machine designed for 
industrial purposes is, under certain restrictions as to the 



EDITORIAL APPENDIX. 60 o 

quality, to yield the greatest amount of its products for the 
motive power consumed, it becomes a subject of prime 
importance to see clearly in what way the work yielded by 
the motive power to the receiver, at its applied point, is 
diminished by the various prejudicial resistances, in its 
transmission through the material elements of the machine 
to the operator, or tool by which the products in question 
are formed. 

The most convenient method for doing this will be to 
place (equation 112, Art. 145) which expresses the relation 
between the work 2]J l of the motive power at the applied 
point and that 2U 2 the work of the operator at the working 

point, with the portion 2TJ + — 2w (y?—v?) which repre- 

y ■ 

sents the work consumed by the prejudicial resistances and 
the inertia, under a form such that the work of each preju- 
dicial resistance shall be separately exhibited, for the pur- 
pose of deducing, from this new form of the equation, the 
influence which each of these has in diminishing the work 
yielded at the applied point and transmitted to the operator. 
To effect this change of form in (equation 112) designate by 
P x the motive . power, and S> the path passed over by its 
point of application in its line of direction between any two 
intervals of time, during which. T 1 may be regarded as vari- 
able both in intensity and direction ; P 2 and S 2 the resistance 
and corresponding path at the working point ; R the various 
prejudicial resistances which, like friction, the stiffness of 
cordage, &c, act with a constant intensity, or are propor- 
tional to P a , and S their path ; w x the weight of the parts the 
centre of gravity of which has changed its level during the 

period considered, and H its path ; and — w (v^—v 1 i )= ha 

-if 
{v*—v*) the half of the difference between the living forces 
or the accumulated work of the material elements in motion, 

of which m ~ — is the mass, during the same period, in 

g 

which the velocity has changed from v, to v 9 . 

Now for an elementary period dt of time, during which 
the forces P, &c, may be regarded as constant, and their 
points of application to have described the elementary paths 
dS t &c, in their lines of direction, (equation 112) will take 
the form, 

lmvdv=^ l dS-nidS-n\dS,±^w l d/i, . . . (A), 



EDITORIAL APPE>T>IX. 

in w : [presses the incre- 

the livi: _ the elemenl emulated 

instant when th 
the mass - 2nd member the -ding 

This 
ion bci _ n die limits t 1 and t t in 

which r chang a 

***«-< =I f* \~- f-^-^f^^.- 
dh (J 



/ 



This equation tB> is the same - nation 112). The 
Bjmb I - lesignating the __ _ rk of the 

::ie same kind; and tha: as / 

the work of each : - P,. sup: sing it 1 be either 

or variable. In eit- - whenever 1' in be 

expressed in terms of S 1 tlie valne SB an be found 

: the me:. s in (B - nd : andgupj 

~ent their mean val s, " :he paths 

bed in their true directions during the interval con- 

ritten under the following form 
for the conveniei. - —'on, 

iw (»/-!•; =P ; S : -r.S-P.S : z:^H (C). 

In this last equation - / w % dh=YfH Art. 60) repr^ 

the v. weight of the parts whose centre of 

gravity has c _ .el during the interval considered, 

and it takes the double sign — . as the path H may be 
er in th. a a contrary direction to that 

in which W alwa - 

Before proc _ lie s the term- i . 

it may be well to remark that the term — I > - not take 
ant the work ezpen ming the 

molecular : - bronghl :he den\ sion, 

the parts of the machi: :.g to 

the rigidity of thee :ns but a small frac- 

tional part of the terior forces vhilst the 

machine open a ..dnuously for some time; as, during 



EDITORIAL APPENDIX. 605 

this time, the tension of the parts, or the molecular resist 
ances remain sensibly the same, and the molecular displace 
raents are for the most part inappreciable, or else very small 
compared with the paths described by the points of applica- 
tion of the other forces. 

This remark, however, does not apply to the expenditure 
of work by the motive power where the operation of the 
machine requires that some of the parts in motion shall be 
brought into contact with others which are either at rest, or 
moving with a slower velocity so as to produce a shock. 
In this case there may be a very appreciable amount of 
living force, or accumulated work destroyed by the shock, 
owing to the constitution of the material of which the parts 
are composed where the shock takes place ; and, if the shocks 
are frequent during the interval considered, and in which 
the other forces continue to act, the accumulated work 
destroyed during this interval may form a large portion of 
the work expended, or to be supplied by the motive power. 
In calculating this amount of accumulated work destroyed, 
we admit what is in fact true in such machines, that the 
interval in which the shock takes place is infinitely small 
compared with the interval in which the other forces act 
continuously, and therefore, in estimating the accumulated 
work destroyed in each shock, that we can leave out of 
account the work of the other forces during this infinitely 
small interval. In this way, considering also that the parts 
where the shock takes place are usually formed of materials 
which undergo an almost inappreciable change of form from 
the shock, and that therefore the mechanical combinations 
of the machine are sensibly the same after the shock as 
before it, we readily see that, to obtain the total expenditure 
of work by the motive power, for any finite interval, we must 
calculate that which is consumed by all the other resistances 
during this interval, and add to this that destroyed by the 
shocks during the same interval, the latter being calculated 
irrespective of the work of the other forces during the short 
duration in which each shock occurs. 

We thus see that, except in some cases where the great 
velocity of the parts in motion may give rise to an appreci- 
able expenditure of work caused by the resistance of the 
medium in which these part.- may be moving, as the air, &C, 
the forces which act upon any machine in motion are the 
motive power; the resistances, such as friction, stiffness of 
cordage, &c., which act cither wiih a constant intensity 
during the motion, or are proportional to the motive power. 



LORIAL APPENDIX. 

the weighl of the parts whose centres of gravity do not remain 
on the same level during this interval; the useful resistance 
arising from the mechanical functions the machine is designed 
for; and the forces of inertia which cither give rise to accu- 
mulated work, or the reverse, as the velocity increases, or 
decreases during the interval considered. 

Resuming equation (C) we obtain, by transposition, 

That is the useful work, or that yielded at the working point 
and which it is generally the object of the machine to make 
as great as possible consistently with the quality of the 
required products, will be the greater as the terms in the 
second member of the equation affected with the negative 
sign are the smaller. 

Taking the term — ItS, it is apparent that all that can be 
done is to endeavor in the case of each machine to give 
such forms, dimensions and velocities to those parts where 
these resistances are developed as will make it the least 
possible. 

With respect to "WH it will entirely disappear from the 
equation when II = o ; in which case the centre of gravity of 
the entire system will remain at the same level ; or else 
only that portion of this term will disappear which belongs 
to those parts of the machine whose centres of gravity either 
remain at rest, as in the case of wheels exactly centered, end- 
less bands and chains, &c. ; or in the case of those pieces 
which receive a motion simply in a horizontal direction. 
This term will also disappear in whole, or in part, in those 
cases where the centre of gravity ascends and descends 
exactly the same vertical distance in the interval correspond- 
ing to the work P,S, ; for during the ascent, as the direction 
of the path H is opp osite to that of the weight W, the work 
consumed will be — "WII, whereas, in the at, it will 

restore the same amount or -f AYII, and the sum of the two 
will therefore be 0. This takes places in the parts of many 
machines, for example in crank arms, and in wheels which 
are not accurately centered ; in both of which cases the 
centre of gravity ascends and descends the same distance 
vertically in the interval corresponding to each revolution 
of these parts whilst in motion; also in those parts of a ma- 
chine, like the saw and its frame in the saw mill, which rise 
and fall alternately the same distance. 

In all of these cases then the useful work P 2 S 3 will not be 



EDITORIAL APPENDIX. 607 

affected by the work due to the weight of the parts in 
question. 

It may be well to observe that the preceding remarks refer 
only to the direct influence of the weight of the parts on the 
amount of useful work ; but whilst directly it may produce 
no effect however great its amount, the weight, indirectly, 
may cause a considerable diminution of this work, by 
increasing the passive resistances and thus the term RS. 
The same holds with regard to the accumulated work, repre- 
sented by the term %mv*, from which a considerable dimi- 
nution may be made in P 2 S 2 if this accumulated work cannot 
be converted into useful work, and thus be made to form a 
portion of P 2 S 2 , when the action of the motive power is either 
withdrawn, or ceases, by variations in its intensity, to yield 
an amount of work which shall suffice for the work consumed 
by the resistances. 

These last remarks naturally lead us to the consideration 
of the two terms -Jm^ 2 , and — Jmv 9 a , or half the living forces, 
or accumulated work at the commencement and end of the 
interval considered. As the machine necessarily starts from 
a state of rest under the action of the motive power P„ it 
follows that \mv*, the accumulated work due to this action 
tends to increase P 2 S 2 , whilst that —\mv* is so much accu- 
mulated in the moving parts by which P 2 S 2 is lessened. 
This diminution of P 2 S 2 is but inconsiderable in comparison 
with the total useful work when the interval in question, and 
during which the machine operates without intermission, is 
great ; also in cases where the velocity attained by the parts 
in motion is inconsiderable, as for example in machines em- 
ployed for raising heavy weights, in which \m r o* will in 
most cases be but a small fraction of the useful work which 
is the product of the weight raised and the vertical height it 
passes through. In this last example we also see the incon- 
veniences which would result from allowing bodies raised by 
machinery to acquire any considerable amount of velocity ; 
or to quit the machine with any acquired velocity, as, in 
this case, the accumulated work generally would be entirely 
lost so far as the required useful effect is concerned. 

Except in the case where the accumulated work \nwl 
can be usefully employed in continuing the motion of the 
machine and gradually bringing it to a state of real when the 
motive power P, has either ceased to net, or lias so far 
decreased in intensity as to be incapable of overcoming the 
resistances, whatever tends to any augmentation el' living 
force should be avoided, for the term which represents this 



... LL APPENDIX. 

being composed of two factors tlie one representing the mass 

of the parrs in motion and the other the square of its wlo- 
ident that the prejudicial resistances Bnch as 
ion on the one hand and the resistance of the air on the 
other will increase as either of these factors is increased, and 
thus a very appreciable amount of this accumulated work 
may be consumed in useless work caused by the very in- 
crease in question. If, moreover, the machine from the nature 
of its operations is one that requires to be brought suddenly 
to a state of rest, any considerable amount of accumulated 
work might bo increase the effects of shocks at the points of 
articulation as to endanger the safety of the pa 

The foregoing remarks apply only to those parts of a ma- 
chine where the direction of motion remains the same whilst 
the machine is in operation. Where any of the parts have 
a reciprocating motion, in which case whilst the part is 
moving in one direction the velocity increases from up to 
a certain limit and then decreases until it again becomes 
at the moment when the change in the direction of motion 
takes place, and so on for each period of change, it will be 
readily seen that where the velocity varies by insensible 
degrees, the accumulated work of these parts for each period 
of change will be and will therefore have no influence on 
the amount P,S. : ot' useful work. 

The avoidance of abrupt changes of velocity in any of the 
parts of a machine is of great importance. The mechanism 
therefore should, as a general rule, be so contrived that there 
shall be the least play possible at the articulations of the 
various parts, and that the articulations shall receive such 
forms as to procure a continuous motion. In cases also 
where any of the parts have a reciprocating motion such 
mechanical contrivances should be used as will cause the 
variations of velocity in these parts, within the range of 
their paths, to take place in a very gradual manner; such 
for examples as what obtains in the cranks and eccentrics 
which are mostly employed to convert the continuous circu- 
lar motion of «'iie part into reciprocating motion in another, 
or the reverse. 

Thei some industrial operations however which are 

performed by Bhocks, as in stamping machines, trip ham- 
iners, &c, and in the* 3 the useful work is due to the 

work developed by the motive power in raising the pestle 
of the stamping machine, or the head of the trip hammer 
through a certain vertical distance from which it again falls 
upon the matter to be acted on, having acquired in its 



EDITORIAL APPENDIX. 609 

descent an amount of living force, or accumulated work due 
to the height through which it has been raised. In such 
cases it is to be noted that, independently of the work due 
to the motive power consumed by the resistances whilst the 
hammer or pestle is kept in motion by the other parts of the 
mechanism, and which is so much uselessly consumed so far 
as the useful work is concerned, there will be a portion of 
the accumulated work in the pestle, or hammer also uselessly 
consumed, arising from the want of perfect rigidity and 
elasticity in the material of which these two pieces are 
usually composed. Besides this, both the pestle and matter 
acted on may and generally do have relative velocities after 
the shock between them, which as they are foreign to the 
purpose of the operation, will also represent an amount of 
accumulated work lost to the useful work. From this we 
may infer that, as a general rule, other industrial modes of 
operating a change of form in matter will be preferable to 
those by shocks, whenever they can be employed ; and that 
such modes are moreover advantageous, as they avoid those 
iars to the entire mechanism which accompany abrupt 
changes in the velocity of any of the parts, and which, by 
loosening the articulations more and more, increase the evil,, 
and ultimately render the machine unfit for service. 

Having examined the influence of all the various hurtful 
resistances brought into action in the motion of machines 
upon the work P& expended by the motive power, and 
pointed out generally how the consumption of the work may 
be lessened, and the useful work to the same extent increased, 
we readily infer that like observations are applicable to the 
term P 3 S 2 the work of the resistance at the working point. 
As the prime object in all industrial operations performed by 
machinery is to produce the greatest result of a certain hind 
for the amount of work expended by the motive power, it 
will be necessary to this end that the velocity, the form, etc., 
of the operator, or tool by which the result sought is to be 
obtained, should be such as will not cause any useless expen- 
diture of work. On this point experiment has shown that 
for certain operators there is a certain velocity of motion 
by which the result produced will be the most advantageous 
both as to the quality and quantity. 

With respect to the work of the motive power itself repre- 
sented by The product P t S, it admits of a maximum value: 
for when the receiver to which P, is applied is at rest, P, will 
act with its greatest intensity, but the velocity then being <> 
the product P.S. will also be"() ; but as the velocity increases 

39 



G10 EDITORIAL APPENDIX. 

after the receiver begins to move the intensity of the action 
of P, upon it decreases, until finally the velocity of the 
applied point may receive Buch a value V that P, will become 

Oj and the product PJS, in this ease will then also be 0. As 
the work PJ3, thus becomes in these two states of the velo- 
city, ii is evident that there is a certain value of the velo- 
city which will make P,S 1 a maximum. To attain this 
maximum the mode of action of the motive power selected 
on each form of receiver to which it is applicable will require 
to be studied, and such an arrangement of its mechanism 
adopted as will prevent any decompositions of the motive 
power that would tend in any maimer to increase the hurt- 
ful resistances and thus diminish the useful work. 

It will be very easy to show that the laws of motion of all 
machines, that is the relations between the times, spaces and 
velocities of the motion of any one of the moving parts are 
implicitly contained in the general equation of living forces 
as applied to machines which has just been discussed. 
Resuming (equation B) with this view, and representing by 
dm any elementary mass in motion whose velocity is v 9 at 
any instant when it has described the path, or space s, if we 
take any other elementary mass dm in a given position and 
denote by v 1 its velocity at the same instant, we shall have 
l\—u^ (<ps), and v l =u 1 (?>$,); in which ys is a purely geome- 
trical function, since, from the connection of the parts of a 
machine, in which any motion given to one part is trans- 
mitted in an invariable manner to the other, the space passed 
over by any one point can always be expressed in terms of 
that passed over by any other assumed at pleasure. 

From the relations v 2 =u 9 (9s), and v^ dt=ds, we obtain 

d?s 
u*((psy=v* and u a du 7 (<p8)*=v i dv i = -r^ds. 

cit 

Substituting these values of v? and v a dv 9 in (equations B and 
A), and letting m still represent the sum of the elementary 
masses as dm, there obtain the two equations 

x[v,d&,±*fwdh. (B 1 ) 



EDITORIAL APPENDIX. 611 

u,du^?n((psy=^m^s) 2< ^ ds = sP^-SltaS - 

2P 2 ^S 2 ±2^A. (A 7 ), 

the first showing the relations between any two states of the 
velocities u 2 and u x for any definite interval, and the second 
for the infinitely small interval dt. Now as the relations 
between the quantities ^S l5 6ZS 2 , &c, or the elementary 
paths described by the points of application of P 1? P 2 , &c, 
and the elementary space ds, from the connection of the 
parts of the machine, can be expressed in functions of s and 
of the constants that determine the relative magnitudes and 
positions of those parts ; and as, moreover, P l3 P 2 , &c, are 
either constant, or vary according to certain laws by which 
they are given in functions of the paths S 1? S 2 , &c, we see 
that all the relations in question are implicitly contained in 
the two preceding equations. 

Let us examine the kinds of motion of which a machine is 
susceptible and the conditions attendant upon them. We 
observe, in the first place, supposing the machine to start 
from a state of rest, that the elementary work Y l dS l of the 
motive power must be greater than that of the resistances 
combined, or Y 1 dS 1 — R<^S— &c. >0, so long as the velocity 
is on the increase. The living force is thus increased at 
each instant by a quantity d (mv' i )=27nvdv, or by an amount 
which is equal to twice the elementary work of the motive 
power and resistances combined ; and this increase will go 
on so long as the elementary work of the motive power is 
greater than that of the resistances. But, from the very 
nature of the question, this increase cannot go on indefinitely, 
for the point of application of the motive power would in the 
end acquire a velocity so great that Y l would exert no effort 
on the receiver, whereas the resistances still act as at the 
commencement, and some of them even increase in intensity 
with the velocity. The living force therefore will, at some 
period of the motion, attain a limit beyond which it will not 
increase, a fact which the operation of all known machines 
confirms, and, having thus reached this state, it -must either 
continue the same during the remainder of the time that the 
machine continues in motion, or else it must commence to 
decrease until the velocity attains some interior limit from 
which it will again commence to increase, and bo on for each 
successive; period of motion during which the action ol the 
forces remains the same. 



EDITORIAL APPENDIX. 

to continue its motion with the velo 
1 at this maximum state of the living force, 
shall then Ik;- 

- ■■-:; S-P, S f ±W«ZH=0; 

and 

»»« f , -m© 1 , =2CPA-BS-l > A±WH)=0; 

3 rhe motion being now uniform the difference 
een the living forces corresponding to any finite inter- 
val of time is 0. Considering the manner in which the parts 
of machines are combined to transmit motion from point to 
. we infer that this condition with respect to the 
of living force, and which constitutes uniform 
an only obtain when the velocities of all the differ- 
ent parts bear a constant ratio to each other. Representing 

. &c, these velocities which are respectively 
■ ■ 
equal to — _, _'_, — — , &c 3 we see that the ratios of <:l<\ 
at 

. will also be constant when those i . Arc 

; that is, this constancy of the ratio of the effective velo- 
- and of the quanta &c, must subsist together 

for all positions of the parts of machines to which they refer; 
but a> the latter, which are the virtual velocities, or ele- 
mentary paths described, depend entirely on the geometrical 

ern the motion of the parts, a little consider;. 

of the various mechanical combinations by which motion is 

■ .ill show that, in order that their ratios shall 

tively remain constant, no pieces having a reciprocat- 

n can enter into the composition of the machine, 

- ■:" such pieces evidently cannot be made to 

.t ratio to the others. This condition it will be 

:clnsivelj to the mechanism of the machii 

letrical conditions by which the parts are connected, 

and - lo with the action of the forces them- 

when the condition of uniform motion is satisfied 

rding to the principle of virtual velocities, an 

equilibrium obtains between the forces which act on the 

machine irrespective of the inertia of the parts. As a gene- 

lition requires that not only must the forces 



EDITORIAL APPEXDIX. G13 

P„ 11, &c, be constant both in intensity and direction and 
act continuously, but that the term \Y<$3 must be sepa- 
rately equal to 0, or the centre of gravity of each part must 
preserve the same level during the motion ; for were this 
not so any piece whose weight is w would evidently impress 
an elementary work represented by ±wdh which would be 
variable in the different positions of the mechanism ; unless 
w, having itself a uniform velocity, formed, as might be the 
case,- a part of the motive power P 1? or of the useful resist- 
ance P 2 . 

It thus appears that to obtain uniform motion not only 
must the mechanism used for transmitting the motion con- 
tain no reciprocating pieces, and therefore consist solely 
of rotating parts, as wheels, &e., and parts moving continu- 
ously in the same direction, as endless bands, and chains, &c. ; 
but that the centres of gravity of these pieces shall remain 
at the same level during the motion, which will require that 
the wheels and other rotating pieces shall be accurately cen- 
tered so as to turn truly about their axes. 

The difficulty of obtaining a strictly uniform motion in 
machines is thus apparent, for it involves conditions in them- 
selves practically unattainable, that is, applied forces acting 
continuously and with a constant intensity and direction, and 
that the ratio of the virtual velocities of the different parts 
should be constant and independent of the positions of the 
mechanism, a condition which requires that the terms (<p*) 
and 2dm(ys) 2 in the preceding equations shall also be con- 
stant for all of these positions. But even were these condi- 
tions satisfied, it can be shown that rigorously speaking a 
machine starting from a state of rest will attain a uniform 
velocity only in a time infinitely great. This will appear 
from geometrical considerations of a very simple character, 
or from the form taken by equation. By the first method, 

let OT, OV be two co-ordinate 
axes, along the one set off the 
abscissas X)t\ Ot'\ &c, to re- 
present the times elapsed from 
the commencement of the mo- 
tion, and the ordinate- tfV, f"r'\ 
&c.j the corresponding veloci- 
ties, the curve ( )r'r'\ &C, will 
give the relation between the times and the velocities. Now, 
from the circumstances of the motion, the increments of the 
velocities will continually decrease, and the curve, from the 
law of continuity, will approach more Dearly to a right lino 




KDITOKIAL APTESDIX. 



M tlu- time increases; having for its assymptote a right line 
parallel to OT, drawn at a distance < h) from it, which is the 
limit the velocity attains when the motion becomes uniform. 
We moreover Bee from the form the curve may assume that 
this limit will be approached more or less rapidly. 

From (equation IT), representing by c the quantity 
', we obtain 

c df ~ c dt~- Ll ds zu ds r ' ds ' 

N<.w. from the preceding discussion, the forces being sup- 
posed to act continuously, and with a constant intensity and 

direction, and the quantities — - 1 ? -^- 2 being constant, the 

ds cis 

function expressed by the second member of this equation 

has its greatest value when t' 9 =0, or when the machine is 

about to move, and that after motion begins it decreases 

more or less rapidly as the velocity increases, until it be- 

comes for a certain finite value of the velocity. Hence it 

follows that the function must be of the following, or some 

equivalent form, 

in which h is essentially positive and a function of v % and 
certain constants, and V is the limit of the velocity in ques- 
tion. We shall therefore obtain from (equation B'), by sub- 
stituting this function for the second member, 



di\ P cdv, 

second member of this last equation, when integrated 
between the limits v„=0, and v t = V, must contain, according 
to the known rules applicable to it, at least one term of the 
form of —a log. (V"—^«) if ^ le exponent n is odd; or 

\ ' "* +1 . it' /' is even; either of which functions will 

me infinite for V— w,=0, or when v 9 attains its limit. 
Prom the conditions requisite to attain uniformity of mo- 
tion in a machine, the advantages attendant upon it, so far 
as it affects the mechanism are self-apparent; not only will 
there be none of that jarring which attends abrupt transi- 

a in the velocity, but, from the manner in which the 






EDITORIAL APPENDIX. 615 

forces act, the strains on all the parts will be equable, and 
the respective form and strength of each can thus be regu- 
lated in accordance with the strain to be brought upon it, 
thus reducing the bulk and weight of each to what is strictly 
requisite for the safety of the machine. But advantages not 
less important than these result from the use of mechanism 
susceptible of uniform motion, owing to the fact that for 
each receiver and operator there is a velocity for the applied 
and working points with which the functions of the machine 
are best performed as respects the products ; and these 
respective velocities can be readily secured in uniform 
motion by a suitable arrangement of the mechanism inter- 
mediate between these two pieces. 

The advantages resulting from uniform motion in machines 
has led to the abandonment of mechanism that necessarily 
causes irregularity of motion, in many processes where the 
character of the operation admits of its being done ; and 
where, from the manner in which the motive power acts on 
the receiver and is transmitted to the operator, parts with a 
reciprocating motion have to be introduced, every possible 
care is taken to so regulate the action of these parts and to 
confine the working velocity within the narrowest limits that 
the character of the operation may seem to demand. Many 
ingenious contrivances have been resorted to for this pur- 
pose, but as they belong to the descriptive part of mechanism 
rather than to the object of this discussion, and, to be under- 
stood, would require diagrams and explanations beyond the 
limits of this work, they can only be here alluded to. There 
is one however of general application, the fly wheel, the 
general theory and application of which to one of the sim- 
plest cases of irregularity are given in (Arts. 75, 76, 265, &c.) 
The functions of this piece are to confine the change of velo- 
city, arising from irregularities caused either by the mechan- 
ism, or the mode of action of the motive power within certain 
limits; absorbing, by the resistance offered by its inertia, 
or accumulating work whilst the motion is accelerated, and 
the work of the motive power is therefore greater than that 
of the other resistances, and then yielding it when the reverse 
obtains; thus performing in machinery like functions to 
those of regulating reservoirs in the distribution of water. 
It should however not be lost sight of that whatever resources 
the fly wheel may offer in this respect they are accompanied 
with drawbacks, inasmuch as the weight of the wheel, its 
bulk and the great velocity with which it is frequently 
required to revolve, add considerably to the prejudicial 



] MTOEIAL APPENDIX. 

friction and the resistance of the air, and time 
a useless consumption of a portion of the work of the 
motive power. Whenever therefore, by a proper adjustment 
of the motive power and the resistances, and a suitable 
arrangement of the mechanism, a sufficient degree of regu- 
larity can be attained for the character of the operation, the 
;' a fly wheel would be injudicious. In cases also where, 
from the functions of the machine, its velocity is at times 
rapidly diminished, or sudden stoppages are requisite, the 
-ily wheel might endanger the safety of the machine, or be 
liable itself to rupture, it should either be left out, or else 
the mass of the material should be concentrated as near as 
practicable around the axis of rotation : thus supplying tho 
requisite energy of the fly wheel by an augmentation of its 
mass. In all other cases the matter should be thrown as far 
from the axis as safety will permit, as the same end will be 
attained with less augmentation of the prejudicial resist- 
ances. 

From this general discussion some idea may be gathered 
of the relations between the work of the power and that of 
the resistances in machines, and of the means by which the 
latter may be so reduced as to secure the greatest amount of 
the former being converted into useful work. It must not 
however be concealed that the problem, as a practical one, 
ats considerable difficulty, and requires, for its satis- 
fy solution, a knowledge of the various operators and 
receivers of power, as to their forms and the best modes of 
their action. This knowledge it is hardly necessary to 
observe must, for the most part, be the result of experiment; 
theory serving to point out the best roads for the experi- 
menter to follow. Both of these have shown that the work 
of the motive power consumed by the resistances, caused by 
the parts through which motion is communicated from the 
receiver to the operator, is but a small fractional part of the 
total work uselessly consumed, whenever the mechanism has 
arranged with proper attention to the functions required 
of it; but that the principal loss takes place at the receiver 
and operator, and this is owing to the difficulty of so arrang- 
ing the receiver that the motive power shall expend upon it 
all it- work without loss from any cause ; and in like manner 
of causing the operator to act in the most advantageous way 
upon the resistance opposed to it. Some of the general con- 
dition- to which these two pieces must be subjected, as to 
uniformity and continuity of action of the motive power and 
the resistances, and the avoidance of jarring and shocks have 






EDITORIAL APPEXDIX. 



617 



been pointed out. as well as the fact that to each correspond? 
a certain velocity by which the greatest amount of useful 
effect will be attained. 

This discussion will make apparent that, comparatively 
speaking, but a small amount of the work due to the motive 
power is expended on the useful resistance, or the matter to 
be operated on. In some of the best contrived receivers, as 
the water wheel, for example, where the motive power can 
be made to act with the greatest regularity, and the receiver 
be brought to as near an approach to uniformity of motion 
as attainable, the quantity of work it is capable of yielding 
seldom exceeds eight tenths of that due to what the water 
expends upon it, under the most careful arrangement of the 
wheel and the velocity of its motion. 



Note (u). 



As an example under this head (Art. 149) equation (115), 
and an illustration of the circumstances attending the attain- 
ment of uniformity of motion Note (t) in machines ; suppose 
^ the axle A carrying two arms B, B, to the 

13 extremities of which two thin rectangular 

[ disks C, C, are attached, their planes pass- 

ing through the axis of rotation, to be put 
in motion by the descent of a weight P, at- 
tached to a cord wound round the axle. 
In this case the resistances to the moving 
force during the acceleration will be 
that of the air acting against the disks and 
the two arms, the inertia of the parts in 
motion, and the friction on the gudgeons 
of the axle. 

Represent by A the sum of the areas of 
the two disks, a the distance of their centres from the axis, 
dm an elementary mass of the machine at the distance r 
from the axis, w the angular velocity of the system, a x the 
radius of the axle measured to the axis of the cord, p the 
radius of the gudgeon, 9 the limiting angle of resistance, 
\ the total length of the cord, I the length of the part 
unwound, w the weight of the unit in length of the cord, \\ 
the total weight of the machine excepting P,. 

From experiment we have for the resistance of the air to 
the motion of the two discs cAi> 9 =cA«V, in which u 




818 1 1 -ITOEIAL APPENDIX. 

expresses fche velocity of the centre of the disk and c a con 
stun! determined by experiment. The resistance offered by 
the inertia of dm during the acceleration of the motion is 

represented (Art 95) equations (72) (73) by dmr -— , in 

(tt 

which *** is the acceleration of the angular velocity in the 

dt 

element of time dt, the resistances offered by the inertia of 
the weight P, and that of the pendant portion of the cord 

represented by wl are, in like maimer, expressed 1 ~ t ~ w a x 

the total pressure upon the gudgeons will evidently be ex- 
pressed by Pj + W— — l — — #, — — , since, during the accele- 
ration of the motion, the resistance of the inertia of the 
weights P, and wl act in an opposite direction to these 
weights. 

In the state bordering upon motion at each instant there 
obtains 

(P 1 + wl)a,=cAu><a> + ldmr^ + F < +wl a^ + 

dt g dt 

Representing by ri 1 the coefficient of w a , by ra a that of — ^1 

at 

and by q 7 the algebraic sum of the other terms, there obtains 



dt * " q* — ?i*<o* 

2qn t 



J f—nW 2qn ° \q-no) n ( *qn f , 

Vs + 1/ 



From this last equation we se that w approaches rapidly the 

limit (/ which it only attains when t=oo . As this limit cor- 
n 

onds to that in which the motion would become uni« 



EDITORIAL APPENDIX. 61fj 

form, it might have been deduced directly from the first of 

these equations; for when w becomes constant— =0, 

dt ' 

n 



ISTote (v). 

Manner of estimating the amount of work consumed by the 
trip hammer. 

The trip hammer is used in forging heavy iron work, 
motion being given to it for this purpose by teeth, termed 




cams, firmly fixed in an axle A, termed the cam shaft, 
around which they are arranged at equal intervals apart. 
The tail of the hammer is furnished with an iron band, the 
upper surface of which receives a suitable form to work 
truly with the surface of the cam whilst the two remain in 
contact during the ascent of the head of the hammer, on the 
same principle as the teeth are fashioned in other Ci 
The interval between the cams is so calculated that each 
cam shall take the band at rest at the point t on the hori- 
zontal line C 2 Cj joining the centres of rotation of the cam 
shaft- and hammer. 

To estimate the work consumed in the play of this ma- 
chine, it. must be observed that it consists of three distinct 
parts; the first is that consumed by the impact or shock ; 
the second that due to the period after the shock, in which 
the cam and tail of the hammer remain in contact ; the third 
that consumed by the cam shaft in the interval between the 
separation of the cam and hammer and the moment when 
the succeeding cam takes the hammer. 

Denote by R 2 the radius of the primitive circle (V of the 
cams; by u 9 the angular velocity of the cam shaft at any 
period of the shock; by p a the radius oi the gudgeon on 



EDITORIAL APPENDIX. 

which the Bhaft revolves ; by p s the limiting angle of resist- 
ance for the surfaces of the gudgeon and its bed; by m an 
elementary mass of the Bhaft ; by r the distance of m from 
0, ; by K, --('/, »„ p„ 9, f»u and r, the corresponding quan- 
tities for the hammer. 

Now if we represent by P the mutual pressure between 
the surfaces of the cam and band at any period of the impact. 
there must be an equilibrium at each instant between P and 
the forces of inertia and the passive resistances developed 
in the play of the machine. Considering the equilibrium 
around the axis of rotation C, of the hammer in the first 
place, we have for the velocity of any element m„ at any 
instant, r x «„ and for the increment of velocity impressed 
upon it by the cam r 1 d^ l ; the force of inertia therefore deve- 
loped by this increment is expressed by 

do), 
m.r. —— - • 
1 * dt ' 

and its moment with respect to the axis C, is 

and the sum of the moments of all the forces of inertia is 
(Arts. 95, 106) 

dt dt 

To obtain the friction on the trunnions of the hammer due 
to P and the resultant of the forces of inertia m 1 r 1 —±, we 

(it 

have for the resultant of the latter (Art. 108) equation {82) 

dt 

in which M represents the mass of the hammer, its handle, 
&c, and Q the distance of its centre of gravity from C, the 
axis of rotation. Now, decomposing this resultant into twe 
components perpendicular and parallel to the line C 2 C, repre- 
senting by a the angle between this line and the one 0,0 
through the centre of gravity of the hammer, &c, we have 
for the perpendicular component 



EDITORIAL APPENDIX. 621 



Hit 



and for the parallel one 

d(o 1 



u MG sin. a. 
at 



The total pressure on the trunnions, from P and the forces 



of inertia, will therefore be 



|/(p + ^MGeo,«J + gMG S in.«) ! . 

As however, in most cases of practice, the angle a is either 0, 
or very small, the value of the quantity under the radical 
may be taken without sensible error 

do) 

P + 4rMG. 

dt 

The equation of equilibrium about the axis C x is therefore 
PE i =-^- l M 1 R 1 i + (P + -$rMG) Pl sin. 9, 
<to x M^' + MGp^in.cp, 

Now with respect to the cam shaft we have, to express the 
sum of the moments of the forces of inertia with respect to 
the axis C 2 , 

»■■ 

As the pressure on the trunnions of this shaft is clue to the 
force P alone, the moment of the friction on them will be 
expressed by P p sin. <p 2 . 

The equation of equilibrium of all the forces with respect 
to C 2 will therefore be 

•^M,E a *=PE,+Pp,8in. *,.... (B). 

Eliminating P between equations (A) and (B) there obtains 



622 EDITORIAL APPENDIX. 



£iw-5 :;: : ; ;( M,.^MG P , 8i ,,)§....(C) 

Che coefficient of — ^ can be written as follows, 
at 

I\„+ p„ sin. <p 2 ,,-.-0 9 , -k T ^ • \ 
-rr '—. (M^rV + MGp, 6111.9,)= 

1 + p Q 6in.(p 9 

iP, sm. 9, \ M.R, / 

r; 

placing Iv for the coefficient of MJU^. Making these sub- 
stitution in equation (C) there obtains 

£iw-emm£ 

•/ dt J dt 

w 2 «i=0 

.-. M 2 E 2 2 («-c l ) 2 )=KM 1 E> 2 

in which ft represents the greatest, and w 2 the least angular 

velocity of the cam ehaft ; and w 1= 0, cj 1 = * » ? the angu- 

1 

iar velocities of the hammer ; since before the impact it is 

at rest, and finally attains the same velocity as the cam has, 
in which, from the circumstance of the mechanism, « X R,= 

From the preceding equation there obtains 
,.» - m * CD) 

Now, as a general rule, the quantities p " ™* ^ 2 ? Pl s ^' ?1 and 

K 2 E x 

p. Bin. p.M<l I n ..-, ., -, 

J — r-j— n4 — are ver J small with respect to unity, and may 



EDITORIAL APPENDIX. 623 

therefore be disregarded, and 'the quantity K will differ but 
very little from unity also. From this it will be seen that 
w 2 will differ the less from ft as M 2 is greater than M r But, 
as the mass of the cam shaft ordinarily very much exceeds 
that of the hammer, we can assume, without liability to any 
great error, that the mean angular velocity of the cam shaft, 
deduced from observing the number of revolutions made by 
it in a given time, is sensibly the arithmetical mean of ft 

and g) 2 . Designating this mean by fy we have Q 1 = fl) '\ 
From this relation and equation (D) there obtains 

^ SQ^+KM,) d 2Q.M, 

2M.+KM, / 2 2M a +KM/ 

From these two relations the living force destroyed by the 
impact can be deduced as follows. Before the impact the 
living force of the cam shaft was ft 2 M 2 R 2 2 ; after the impact, 
as the point of contact of the cam and band moved with the 
same velocity, the living force of the whole machine is 

< M 2 K 2 2 + «/ M^ 2 = co 2 2 E 2 2 (M 2 + M,). 

The living force destroyed therefore is expressed by 

l2 2 M 2 K 2 2 -o) 3 2 E 2 2 (M 2 + M 1 ); 

or, substituting for w 2 from equation (D), by 

2 . E .j M M a '(M,+M.) 
aK < {*• (M.+KM,)" 

2 2 ( (M. + XM,)' 

finally, substituting for ft and w 2 their values in Q,, there 
obtains 

M 

It is now readily seen, from the form of this last expression 
for the loss of living force by the impact, that, Bince K tnaj 



024 EDITOBIAL AIM'K.NDIX. 

ly equal to unity, the numerical value 
of thi ' will depend npon the ratio — -L. Taking 

M M the value of the expression becoim 1 ;_'-': and 

for M - f- it becomes Q/Jn K..\ Therefore between these 
limits the difference is \ only of the living force lost under 
the supposition of M,=oc . 

In the ordinary arrangement of this machine it rarely 
occurs thai M, is not less than xV^r,. Assuming this as the 
limit, and substituting in the preceding expression 10M, for 
M , there obtains for the required lose of living force 
- 9977Q > *M 1 'Ko'. It is therefore seen that, in all usual c 
M.. may be assumed as infinite without causing any notice- 
able error in the result. 

To estimate the accumulated" work expended by the cam 
shaft for each shock, ft, <*\ and ft, being the same as in the 
preceding expression, this work is expressed by 

4fyM,M,R,'K 
(n_ftfHR = 2Mj+KMi ■ 

As the cam shaft expends this amount of* accumulated work 
ch impact, a quantity of work equal to the half of this 
must be yielded by the motive power at each impact, or 

2n l 'M a M 1 R a a K 
2M.+KM, * 

If therefore there are N cams on the shaft, and it makes n 
revolutions in one minute, then the work consumed by the 
number of shocks in one second will be expressed by 

Kn 2n i a M > M 1 R, a K 
GO 2M.+KM, ( ) * 

Tliis then is the work consumed by the impact in one second 
to'- the first period of the play of the machine; and it has 
been calculated according to what was laid down in Note (t) 
on the subjecl of shocks, by disregarding the work of the 
other l'or< appreciable during the short interval of the 

imp. 

ii late now the work expended during the second 

period, or whilst the cam and band are in contact after the 

C,G any ] -ition of the line C,G, during this 



EDITORIAL APPENDIX. 625 

period, making an angle G^G^a with its position when 
the hammer is at rest. Represent by P x the normal pressure 
at the surface of contact of the cam and band which will 
balance all the resistances developed in the motion of the 
hammer, leaving out of consideration that of inertia, as the 
change of velocity between the end of the impact and when 
the cam disengages from the band is so small that the living 
force due to this interval may be neglected in comparison 
with the work of the other forces ; by W x the weight of the 
hammer, its handle, &c. 

When the line GC, is in the position Gfi x , the line Gf will 
De in that G& making the angle tG 1 t l =a with its original 
position. The force P, acting at t x in this position and per- 
pendicular to the line t 1 G 1 — since the surface of the band 
produced passes through the axis C n the surface of the cam 
being an epicycloid — has for its vertical and horizontal com- 
ponents P, cos. a and P x sin. a. The pressure on the trun- 
nions of the hammer, which is the resultant of P, and W\, 
therefore will be expressed by 



V{W, + P, cos. a) 2 + P t 2 sin. 2 a ; 

and since the first term of the radical is in all cases greater 
than the second, the value of the radical itself may be 
expressed by (Note B) 

7(W 1 + P 1 cos. a) + / 3 P 1 sin. a. 

The equation of equilibrium between P, and the other forces 
will therefore be 

PJR^W^cos. 0+a)+ jr(W 1 +P 1 cos.a)+/3P 1 sin.a^ 1 Bin.<p 1 . 

The moment of the friction at the point /?„ due to P, with 
respect to the point C„ in this case from the form of the cam 
and band, being 0. 

As the pressure P, varies with the angle a, we ran only 
obtain its mean value by first finding its quantity of work 
for the angle a=a, described whilst the cam ami band are in 
contact. Multiplying the lasl equation by da, and then 
integrating between a=0 and oc=a, there obtains 



fl\n i da=W 1 G{sm.(a + a 1 )-^uua\ -f- [yW^+yP* sin.a,- 

«=0 j8PmC08.a l H 3P m ( Pl 8in.9,j 

40 



P , the mean value of P, or tne constant 
t. which multiplied by Ka,, the 
pat h d by the point of application, will zive the 

aIil( ,u >rk of The variable pressure P, : 

I introducing this mean value into the term 01 the 
d that lv - a the moment of the fric- 
tion on the trunnions, as this will not produce any sensible 
error in the results. . 

\ - or that the quantity G|e - 

al hei Jit through which the centre of g: 
the hammer. & . - B during the period in question, and 
that P m P a. is the work of the mean force ; calling this ver- 
tical height h. and substituting the work of the mean for 
that of the variable force in the" last equation; there obtains 

P«Ro 1 = 1 W W+ ... -: Pm sin. a,— PP. cos. a : + 

PP4p 1 siii.f . 

. p Tr^-;AV : a, : sin.r : (E 

" m S .:,.-■:? .in.a.- 3 I-o>.a : j 0,5111. 5, 

If we now multiply the second member of equation (E 
R,a, we shall obtain* the approximate value of the work of 
the variable force P, during the period in question ; or the 
ralue fP K as determined from equation 

T i find now the work that the motive power must supply 
be cam shaft for this expenditure P^P^a, due to the 
given to the hammer during the period inquee 
and also that arising from the resistances developed b; 

:' the cam shaft itself during this period, represent 
which, acting at a distance E, from the as 
of the cam shaft, will balance all the resistances aroun 

the cam shaft and its fixtures : by 6 any 

e'descril the cam shaft during the period con- 

. and p the limiting angk- stance at the point 

:' the cam and band. 

Th< re on the trunnions of the cam shaft is evidently 

expressed by 

W.+P.-P.; 

and the equation that expresses the work of P, for the ele- 
mentary angle ' ; 



EDITORIAL APPENDIX. 



627 



VJW=I> m Rje + *j> m ^l^ tan. , K^+ 

Sow. representing by P. the mean value of P., and substi- 
tutmg it for P m the last term of the second member of tins 
equation which may be done without causing any sensible 

rcnan^S^r^/p^^/ 0111 *»«&*£ <* S 
mecnamsm that K^B^; and integrating this equation 

between the limits 6 = and 0=0, = ^ ; there obtains, to 
express the total work of P, for the angle 0„ 

0=0 -^i^ 2 

(W s + P H -P m ) Pa sin.p^B, 

(Art mW^ consum , ed by the friction of the axles in equation (251) 
(Art 220) that which is expended on the teeth in contact whilst the are r ■d, d 
described is represented by the term of the equation ^ W 

iV'/l-f- — )sin. 0P 2 a 2 ^,. 

to f5,5. W V UP? + if- e ! 2=r2 ' "^ P2 aCtS at the P° int of contact ™<* normal 
to the surface, this term, modified to suit the supposition, becomes 

W 1 + M sin. <p Z!_ r 2V , = £P a /I±2\ r rfi tan. <p= 
\ r x J cos.tf \ r, / 

\ r t r t / 2 
Dividing this last expression by r 2 i/>, there obtains, 

\ n r 2 / 2 

as the value of a mean or constant force which applied tangentialh to the 
circumference having the radius r, will expend, whilst the point of application 
describes the arc r^, the same quantity of work as that consumed bj the fric- 
tion of the teeth in contact, whilst this arc is described. In this expression 
the value of P 2 is less than the true value. 

The foregoing is the theorem of M. Ponoelel referred to on mum kII 
Authors Preface. The direct manner of deducing i. is found on 
Wavier. Resume des Logons, &c. Troisieme Partie Paris I 



1 I'll OKI AL APPENDIX. 

!•.., i 'I^ii'.'^^tan.^+OV-P^.sm.?, 

P R.R. a (F) 

TR^-PjBin.?,. 

K, 

The work therefore that the motive power must supply to 
the cam Bhaft during this period is found by multiplying the 

second member of equation (F) by Rjfl^R,-^- 1 or the path 

d over by the point of application of the mean force P„ 
during this period. 

Representing in like manner, by -^- the number of times 

the hammer is raised per second, the quantity of work that 
the motive power must supply for this expenditure will be 
expressed by 

60 " 3 ' 60 " ' R, w 

During the last period, or whilst the hammer is down, the 
motive power will only have to supply the expenditure of 
work caused by the friction on the trunnions of the cam 
shaft, arising from the weight of this shaft and its fixtures 
and the power; any accumulation of work in this shaft 
during this period being neglected as small in amount. 
Representing byj>=N fhe number of cams on the shaft, their 
ace apart on the primitive circumference whose radius 

2*K 

is B is evidently -j and, as the arc described on this 

circumterence whilst the cam shaft and hammer are engaged 

2tR„ 

is R.0.. that described whilst the hammer is down is -— 

P 
Calling T p the power which acting at the distance R, 
will balance the friction arising from the weight W, of the 
cam Bhaft and fixtures and P , The value of P p will be found 
according to the conditions stated as follows, 

P p R,=(W,+P J ,)f>,sin.<p r 

W.p.sin.?, 



P 



9 



p R 3 — p 2 sin. a, 
The work of P D is 

R^/SMEt, 



K,/2cK, V 






EDITORIAL APPENDIX. 629 

as the path passed over by its point of application is evi- 
dentl y the arc ^ I - — - 2 — E.a, I . 

The work which the motive power must supply therefore 
per second during this last period is expressed by 

W^nSjT _I W ■•■■®- 

By taking the sum of the quantities expressed by the 
formulas (1), (2), and (3) there obtains 



60 



^M^KMT + " 3 1+ P %\~T ll )\ 



to express the total work that the motive power must yield 
to the cam shaft per second to supply the work consumed 
by all the resistances. 

Tnat consumed by the useful resistances, which consist of 
half the living force transmitted to the hammer and the 
work consumed in raising the centre of gravity of the ham- 
mer, &c, through the vertical height h is represented by 

From the preceding expressions, it is easy to deduce the 
work which must be expended in producing a given deptli 
of indentation by the hammer upon the metal when brought 
to a given state of heat. For this purpose, we observe that 
to half the living force acquired by the hammer there cor- 
responds a certain amount of work, estimated in terms of 
the weight of the hammer and a certain height A, to which 
its centre of gravity has been raised, and expressed by 

the total work therefore expended by the hammer in 
indenting the metal is expressed by W,//, + W x h ; since, from 
the state of the metal the molecules which are displaced by 
the impact acquire velocities which are Dot appreciable from 
their smallness ; the resistances therefore offered by the 
metal to indentation may be regarded as independent of the 



0:;0 rOBIAL APPENDIX. 

velocity and, from the laws of the penetration of solids intc 
different media, proportional Biinply to the area of the inden- 
tation. Representing then by a and b the sides of the area 
of the indentation, supposed rectangular, at the surface of 
the metal impinged on, d the depth of the indentation, and 
the constant ratio of the resistance and the area of the 
indentation, the following relation obtains between the work 

uded by the hammer in its fall and that offered by the 

ance of the metal 

an equation from which C may be determined by experi- 
ment in any particular case. 

It will be readily seen that the preceding expressions will 
be rendered applicable to the cases where the cam catches 
the hammer on the same side of its axis of rotation as its 

centre of gravity, by writing — — l MG for + — S^-MG, and 

Cut at 

moreover in this case when P— - — L MG=0, there will be no 

dt 

shock on the trunnions (Arts. 108, 109), and there then 

obtains, to find the point where the cam should catch the 

hammer corresponding to this case, 

p MG 

• Morin, Suite des Xouvelles Experiences stir le Frottenunt, p. 67. Paris, 1836. 



APPENDIX. 



NOTE A. 
b 

Theoeem. — The definite integral j fxdx is the limit of the sums of thi 

a 
valites severally assumed by the 'product fx . A x, as x is made to vary by 
successive equal increments of Ax, from a to b, and as each such equal 
increment is continually and infinitely diminished, and their number there- 
fore continually and infinitely increased. 

To prove this, let the general integral be represented by Fx ; let us sup- 
pose that/3 does not become infinite for any value of x between a and b, 
and let any two such values be x and x + Ax ; therefore, by Taylor's the- 
orem, F (x + Ax) = Fx + Axfx + (Az) J + XM, where the exponent 1 + % is 
given to the third term of the expansion instead of the exponent 2, that the 

case may be included in which the second differential coefficient of Fz, ~ , 

dx 

is infinite, and in which the exponent of A a; in that term is therefore a 

fraction less than 2. 

Let the difference between a and b be divided into n equal parts ; and 

let each be represented by Ac, so that 

b — a 

= Ax. 

n 

Giving to x, then, the successive values a, a+ Ax, a + 2 Ax . . a + (n — 1) 

Ax, and adding, 

F(a+nAx)=~Fa + AxX l f{a + (n— 1) Aa;} + (A<r) 1 + XrM„ 

.-. F&— Ya=AxX i 'f{a + (n— l)Ar} + (Ax)' + X2M n . 

Now none of the values of M are infinite, since for none of these values is 

fx infinite. If, therefore, M be the greatest of these values, thou is J M, lea 

than tiM : and therefore 

FJ — Fa — AxX 1 f{a + (n— 1) Ax} < (b— a) M (Ar)\. 

The difference of the definite integral Yb — Fa, and the sum S,"(A»)/ a 

(n — 1) Ax} is always, then' fore, less than (b — a) M (A.r)\. Now M '^jiniff, 

and (b — a) is given, and as n is increased Ar is diminished continually ; 

and therefore (Ax)\ is diminished continually, ?. being positive. 

Thus by increasing n indefinitely, the difference of the definite integral 



APPENDIX. 

and the sura may be diminished indefinitely, and therefore, in the limit, the 
definite Integra] is equal to the sum (i. c.) 

Yb — Fa = limit £,"(Aii) .f{a + (n-l) Ax} ; 
or, interpreting this formula, Yb — Fa is the sum of the values of Ax .fx, 
when x i> made to pass by infinitesimal increments, each represented by 
Aj, from a to b. 



NOTE B. 
Poxoelet's Fiest Theorem. 

* The values of a and b in the radical V a- + b* being linear and rational, 
let it be required to determine the values of two indeterminate quantities 
a and 3, such that the errors which result from assuming y 7 a 3 + b 2 = a<z + 3&, 

through a given range of the values of the ratio ( 5 ), may be the least pos- 
sible in reference to the true value of the radical ; or that aa " r ' . 

W + 5 3 

art + 36> 
or .— — =_ — 1 5 may be the least possible in respect to all that range of 

values which this formula may be made to assume between two given 

a . a 

extreme values of the ratio 7. Let these extreme values of the ratio t 

be represented by cot. c, and cot. c 2 , and any other value by cot. 4. Sub- 

a . 

stituting cot. 4 for 7 in the preceding formula, and observing thaty a 2 -|-& s 

= -\/b*coi. , ^ + b' i =b cosec. 4, also that aa + pb = a5 cot. 4 + 3&= (a cos. ^.+3 
Bin. 4-)5 cosec. 4, the corresponding error is represented by 

a cos. 4. + 3 sin. 4 — 1 (1); 

which expression is evidently a maximum for that value ^ 3 of xp which is 
determined by the equation 

a 

cot. u/ 3 =3 ( 2 ); 

so that its maximum value is 

V^Tt'—l (3). 

Moreover, the function admits of no other maximum value, nor of any 
minimum value. The values of a and 3 being arbitrary, let them be 

assumed to be such that- or cot. o 3 may be less than cot. *h % and greater 

* The method of this investigation is not the same as that adopted by M, 
Ponceh-t ; the principle is the same. 



poncelet's theorem. 633 

than cot. 4/ 2 . Now, so long as all the values of the error (formula 1) 
remain positive, between the proposed limits, they are all manifestly di- 
minished by diminishing a and j3 ; but when by this diminution the error 
is at length rendered negative in respect to one or both of the extreme 
values 4*!, or 4 2 of 4- 5 and to others adjacent to them, then do these nega- 
tive errors continually increase, as a and (5 are yet farther diminished, 
whilst the positive maximum error (formula 3) continually diminishes, 
Now the most favorable condition, in respect to the whole range of the 
errors between the proposed limits of variation, will manifestly be attained 
when, by thus diminishing the positive and thereby increasing the negative 
errors, the greatest positive error is rendered equal to each of the two 
negative errors ; a condition which will be found to be consistent witn 
that before made in respect to the arbitrary values of a and j3, and which 
supposes that the values of the error (formula 1) corresponding to the 
values 4^ and 4 2 are each equal, when taken negatively, to the maximum 
error represented by formula 3, or that the constants a and |3 are taken 
so as to satisfy the two following equations. 

1— (a cos. -^i+jS sin. ■*■,)= Va?+j?— 1. 

1— (a cos. "^i+jS sin. "*"i)=l — (a cos. Wz + fi sin. ^). 

The last equation gives us by reduction 

cos. -$■(•¥■ 1 — ipj) 
a cos. ¥ 1 + /3 sin. *i=P sin . K g i+ Va) > 

and a = j3 cot. i(V l + , & a ). 

•Substituting these values in the first equation, and reducing, 

2 sin. JOPi+^s) = sin. K^i + ^») / 4V 

P ~~l+cos. £(■¥■, + ■¥■,) cos. 2 i(-*- x — ¥ 2 ) * ' ' * w ' 

_ 2 cos. £(■*•,+¥,) _ cos. j(¥,—¥ a ) ( . 

•'* a ~ 1 + cos. K% — %)~ cos. «*(*, - *0 W< 

These values of a and /3 give for the maximum error (formula 3) the ex- 
pression 

tan. a K*i— **) ( 6 )- 

Thus, then, it appears that the value of the radical y/aF+V is represented, 

in respect to all those values of t- which are included betweon the limits 

cot. ^ and cot. ^ 2 , by the formula 

„ cos .l(% + %) , sin. K^. + Xp 2 ) ( V) 

cos. 2 *(*, - * 2 ) + cos. 3 ±(*i - ^ Wl 

with a degree of approximation which is determined by the value of 
tan. 9 *(*■,— %). 

If in the proposed radical the value of a admits ol being moreased in- 
finitely in respect to 5, or the value of & infinitely diminished in reaped u> 
a, then cot. ^F, = infinity ; therefore 1\ = 0. In this oase the formula oi 
approximation becomes 



634 



APPENDIX. 



a{\— tan. i i%) + 2btan.i% (8); 

and the maximum error 

tan. 2 i% (9). 

If the values of a and 5 are wholly unlimited, so that a may be infinitely 
small <»r infinitely great as compared with 6, then cot. M^ = infinity, 
cot. % = ; 

therefore ¥,=0, ¥«=«. Substituting these values, the formula of approx- 

iination becomes 

•8284a + -82845 (10); 

and the maximum error 

•1716, or jth nearly. 

If 6 is essentially less than a, but may be of any value less than it, so 

a 

that j- is always greater than unity, but may be infinite, then cot. ^ = in- 
finity, cot. ip 2 =l ; therefore ^,=0, &=t. Substituting these values in the 
formula of approximation, and reducing, it becomes 

•96046a + -397835 (11); 

and the maximum error 

•03945, or ^th nearly. 

It is in its application to this case that the formula has been employed in 
the preceding pages of this work. 

The following table, calculated by M. Gosselin, contains the values of 
the coefficients a and j3 for a series of values of the inferior limit cot. «^„ the 
superior limit being in every case infinity. 



Relation of a to b. 


3 +s 


Value of a. 


a and b any ) 
whatever ) 





082840 


a> b 


l 


096046 


a > 26 


2 


0-98592 


a > 36 


3 


0-99350 


a > 46 


4 


0-99625 


a> 56 


5 


0-99757 


a > 66 


6 


0-99826 


a> 76 


7 


099875 


a > 86 


8 


099905 


a > 96 


9 


0-99930 


a> 106 


10 


099935 



Value of/?, 



Maximum Error. 



0-82840 

0-39783 
0-23270 
0-16123 
0-12260 
0-09878 
0-08261 
0-07098 
0-06220 
005535 
0-04984 



0-17160 or $ 

0-03954 or ^ 
0-01408 or -A- 
0-00650 or y ] T 
2irs 



0-00375 or 



0-00174 or T ^ 
000125 or ^ 
0-00095 or T ^ 
0-00070 or J ^ ll 
000065 or -^ 



Approximate Value 
of * «*+&». 



0-8284 (a + 6) 

•96046a + -397836 
•98592a + -232706 
•99350a + -161236 
•99625a + -122606 
•99757a + -098786 
•99826a+ 082616 
•99875a + -070986 
•99905a + -062206 
•99930a + -055356 
•99935a + -049846 



POKCELET S SECOND THEOREM. 63 J 

Poncelet's Second Theoeem. 

To approximate to the value of Va?-V, let m — Qh be the formula of 
approximation, then will the relative error he represented by 



v^ — ' orbyl - 



(±5 

.Now, let it be observed that o? being essentially greater than 5 a , -> 1 ; 

a 

let ^, therefore, be represented by cosec. 4,, then will the relative error be 

, , , (a cosec. 4— j3) 
represented by 1— , or by 

r cosec. 2 4<— 1 

1— a sec.4/+j8 tan. 4, (12), 

which function attains its maximum when sin. 4 = - Substituting thia 

0, 

value in the preceding formula, and observing that —a. S30 ^ + 18 tan. 4, = 

, . , (-?) 

— sec.4, (a— 3 sin.4>)=— _ = — -Va 2 — j3 2 , we obtain fo* the maximum 

error the expression 

1 — Vo?— 3* (13), 

Assuming 4^ and 4» 2 to represent the values of 4/, correspond!^ to the 

greatest and least values of -, and observing that in this case, as in the 

preceding, the values of a and j3, which satisfy the condition? of the 
question, are those which render the values of the error corresponding to 
these limits equal, when taken with contrary signs, to the maximum error, 
we have 

— l+asec.^ — j3tan.^, = l — Va. 7 — 3" .... (14). 
1 — asec.4i + |3tan.4/j=l — a sec. 4* + 3 tan | 2 . . . . (15). 
The latter equation gives, by reduction, 

_„ cos. j(4,— 4v) n . 

a, = 3— 7-7- r-\ .... (loy- 

H sin. £ (4m + 4-,) 

a «» — «*i cos-'H*!--* *) _ , I _ A , oo s.frco8.fr 
a — P ~ P (sin. 8 £>,+>.) (""' 5 sm.'Kfr + fr> 
And a sec. fr -/3 tan. fr = |3 cot. £ (fr +fr) • • • « v 1 ?)- 
Substituting these values in equation (1-i), and solving in respect to |! 






63G APPENDIX. 



3 =_ - -) (18). 



COS. i (r i -r - J - S CM. ft COS. ip, 

2 c< >s. ! (fa — _ 

a = . . i 

- 

The maximum error Lb represented by the formula 



(19). 



x 2 Vcos. xp t cos, il, ..(20) 

cos. i (e i + 1//,) + Vcos. Cj cos. & 
These formula? vrill be adapted to logarithmic calculation, if we assume 

i fo +£)=¥' and cos ^(^ — S ') = cosec. ¥ 2 ; we shall thus obtain from 

sm. 

equations (16) and (17) a = ,3 cosec. ¥j, 4V — 3 2 = 3 cot. ¥„ and a sec. t>, 
— j3 tan. ^i = 3 cot. ¥, ; therefore, by equation (14), 

2 2 sin. ¥, sin ¥/ 



(21). 



^ cot. ¥, + cot. ^ 2 sin. (¥, + *,) 
2 cosec. X K> 2 sin. X P, 

a = cot. *, -r cot. * 2 ~~ sin. (*, + ^ 2 ) . 

, r . sin. (¥, — ¥,) 

Maximum errror = - — )-;- -r-f ^ ns 

sin. ('P, + * s ) (22). 

The form under which this theorem has been given by M. Poncelet is 
dhferent from the above. Assuming, as in the previous case, the limiting 

values of f to be represented by cot. £, and cot. C„ and proceeding bv a 


geometrical method of investigation, he has shown that if we assume 

tan. ^1 = cos. co„ tan. t 2 = cos. wj, uj+u, = 2y„ u t — u, = 25, and cos. y, = 



cos. b 



lL\ then 



2 cos. y, 2 cos.--,, . sm. (y, y,) 

a= - — ; — ., 3=- — r-± , maximum error = - — / — .. 

Sin. (yi+y,) ' Sin. (y, + y 2 ) COS. 6 Sin. (y, + y,) 

If the least possible value of a be l T l „&, and its greatest possible value 
be infinite as compared with &, M. Poncelet has shown the formula of 
approximation to become 



Va* — b* = 1-I319c7 — 0-726365 (23), 

with a possible error of 0-1319 or 4 nearly. 

If the least possible value of a be 2&, and its greatest possible value 
infinite c< >mpared with b ; then 

i/^Iy = i-oi 8623a — 0-2 72944& (24), 

with a possible error of -0186 or 5 ^d nearly. 



ON THE ROLLING OF SHIPS. 637 

NOTE C. 

Ox the Rolling of Ships. 

(First published by the Author in the Transactions of the Royal Society 
for 1850, Fart II) 
Let a body be conceived to float, acted upon by no other forces than its 
weight W, and the upward pressure of the water (equal to its weight) ; 
which forces may be conceived to be applied respectively to the centre of 
gravity of the body and to the centre of gravity of the displaced fluid ; 
and let it be supposed to be subjected to the action of a third force whose 
direction is parallel to the surface of the fluid. Let AH^ represent the ver- 
tical displacement of the centre of gravity of the body thereby produced*, 
and AH 2 that of the centre of gravity of its immersed part. Let more- 
over the volume of the immersed part be conceived to remain unaltered t 
whilst the body is in the act of displacement. If each centre of gravity 
be assumed to ascend, the work of the weight of the body will be repre- 
sented by — W.aH,, and that of the upward pressure of the fluid by + 
W.aH 2 , the negative sign being taken in the former case because the force 
acts in a direction opposite to that in which the point of application is 
moved, and the positive sign in the latter, because it acts in the same direc- 
tion, so that the aggregate work Sw 2 (see equation 1, p. 122.) of the forces 
which constituted the equilibrium of the body in the state from which it 
has been disturbed is represented by 

— W.aH. + W.aH,.} 
Moreover, the system put in motion includes, with the floating body, the 
particles of the fluid displaced by it as it changes its position, so that if 
the weight of any element of the floating body be represented by w„ and 
of the fluid by w 2 , and if their velocities be v l and v 2 , the whole vis viva is 
represented by 

* "When a floating body is so made to incline from any one position into any 
other as that the volume of fluid displaced by it may in the one position be 
equal to that in the other, its centre of gravity is also vertically displaced ; 
for if this be not the case, the perpendicular distance of the centre of gravity 
of the body from its plane of flotation must remain unchanged, and the form 
of that portion of its surface, which is subject to immersion, must be determined 
geometrically by this condition; but by the supposition the form of the body 
is undetermined. It is remarkable what currency has been given to the error, 
that whilst a vessel is rolling or pitching, its centre of gravity remains I 
I should not otherwise have thought this not.' necessary. 

| This supposition is onh r approximately line. 

X If the centre of gravity of the body or of the displaced Said descends (» 
property which will be found to characterise a Large class of vessels), -All, iu 
the one case, and All, in the other, will of course tike the negative sign. 



PBHDIX. 

and we have by equation I -- 

I— W(AH,— AB^=ix« l «5+^2«^ .... (25). 

In the extreme position into which the body is made to roll and it 
which 1 ' 

( =W.,>II ; -aIIj-^ ? ^ . . . . (26). 

or if the inertia of the displaced fluid be neglected. 

D *; = W.,aH, — AH,) 27). 

Wksnee it foUamthat the work necessary to incline a floating body 
through any given angle is equal to that necessary to raise it bodily through 
the dine/ence of the vertical displacements of its centre 
mersed part ; so that other thing* 
the same, that ship is the most stable the product of whose weight by this 
■net is the greatest. 
In the case in which the centre of gravity of the displaced fluid descends, 
the nan of the displacements is to be taken instead of the difference. 
This conclusion is nevertheless in error in the following respects : — 
1st. It supposes that throughout the motion the -weight of the displaced 
fluid remains equal to that of the floating body, which equality cannot 
accurately have been preserved by reason of the inertia of the body and 
of the displaced fluid.* 

From this cause there cannot but result small vertical oscillations of the 
body about those positions which, whilst it is in the act of inclininj. 

] to this equality, which oscillations are independent of its principal 
oscillation. 

2ndly. It involves the hypothesis of absolute rigidity in the floating 
BO that the motion of every part and its vis riva may eeaSe :. 
when the principal oscillation terminates. The frame of a ship and its 
se. however. I by reason of this • .ere cannot 



* The motion of the centre of gravity of the body being the same as though 
all the disturbing forces were applied directly to it, it follows, that no elevation 
of this point is caused in the beginning of the motion, by the application of a 
tal disturbing force, or by a horizontal displacement of the weight of 
the body, which, if it be a ship, may be effected by moving its ballast. The 
>f rotation thereby produced takes place therefore, in the first instance, 
about the centre of gravity, but it cannot so take place without destroying the 
equality of the weight of the displaced fluid to that of the m this 

inequality there results a vertical motion of the centre of gravity, and anothei 
axis of rotation. 



ON THE ROLLING OF SHIPS. 639 

but result oscillations, which are independent of, and mny not synchro- 
nise with, the principal oscillation of the ship as she rolls, so that the vis 
viva of every part cannot be assumed to cease and determine at one and 
the same instant, as it has been supposed to do. 

3rdly. No account has been taken of the work expended in communi- 
cating motion to the displaced fluid, measured by half its vis viva and 

represented by the term —^w&l in equation 26. 

From a careful consideration of these causes of error, the author was 
led to conclude that they would not affect that practical application of the 
formula which he had principally in view in investigating it, especially as 
in certain respects they tended to neutralise one another. The question 
appeared, however, of sufficient importance to be subjected to the test of 
experiment, and on his application, the Lords Commissioners of the Admi- 
ralty were pleased to direct that such experiments should be made in Her 
Majesty's Dockyard at Portsmouth, and Mr. FnsrcnAM, the eminent Master 
Shipwright of that dockyard, and Mr. Kawson, were land enough to 
undertake them. 

These experiments extended beyond the object originally contemplated 
by him ; and they claim to rank as authentic and important contributions 
to the science of naval construction, whether regard be had to the prac- 
tical importance of the question under discussion, the care and labor 
bestowed upon them, or the many expedients by which these gentlemen 
succeeded in giving to them an accuracy hitherto unknown in experiments 
of this kind. 

That it might be determined experimentally whether the work which 
must be done upon a floating body to incline it through a given angle be 
that represented by equation 27, it was necessary to do upon such a body 
an amount of work which could be measured ; and it was further neces- 
sary to ascertain what were the elevations of the centres of gravity of the 
body and of its immersed part thus produced, and then to see whether 
the amount of work done upon the body equalled the difference of these 
elevations multiplied by its weight. 

To effect this, the author proposed that a vessel should be constructed 
of a simple geometrical form, such that the place of the centre of gravity 
of its immersed part might readily be determined in every position into 
which it might be inclined, that of its plane of flotation being supposed to 
be known ; and that a mast should be fixed to it, and a long yard to this 
mast, and that when the body floated in a vertical position a weight 
suspended from one extremity of the yard should raddenly be allowed to 
act upon it causing it to roll over; that the position into which it thus 
rolled should be ascertained, together with the corresponding elevations 
of its centre of gravity and the centre of gravity o\~ its immersed part, 
and the vertical descent of the weight suspended from the extremity of 
its arm. The product of this vertical descenl by the weight suspended 



APPENDIX. 

from the arm ought then, by the formula, to l>e found nearly equal to the 
difference of the elevations of the two centres of gravity multiplied by 
the weight of the body; and this was the test to which it was proposed 
that the formula Bhould be subjected, with a view to its adoption by prac- 
tical men as a principle of naval construction. 

To give to the deflecting weight that instantaneous action on the ex- 
tremity of the arm which was necessary to the accuracy of the experiment, 
a string was in the first place to be affixed to it and attached to a point 
vertically above, in the ceiling. When the deflecting weight was first 
applied this string would sustain its pressure, but this might be thrown 
at once upon the extremity of the arm by cutting it. A transverse sec- 
tion of the vessel, with its mast and arm, was to be plotted on a large 
scale on a hoard, and the extreme position into which the vessel rolled 
being by some means observed, the water-line corresponding to this 
position was to be drawn. The position of the yard, in respect to the 
surface of the water in that position, would then be known, and the vertical 
descent of the deflecting weight could be measured, and al^o the vertical 
ascent of the centre of gravity of the immersed part or displacement. 

To determine the position of the centre of gravity of the vessel, it was 
to be allowed to rest in an inclined position under the action of the deflect- 
ing weight; and the water-line corresponding to this position being drawn 
on the board, the corresponding position of the deflecting weight and of 
the centre of gravity of the immersion w T ere thence to be determined. 
The determination of the position of the vertical passing through the 
centre of gravity of the body would thus become an elementary question 
of statics; and the intersection of this line, with that about which the 
section was symmetrical, would mark the position of the centre of gravity. 
This determination might be verified by a second similar experiment with 
a different deflecting weight. 

These suggestions received a great development at the hands of Mr. 
Rawson, and he adopted many new and ingenious expedients in carrying 
them out. Among these, that by which the position of the water-line 
was determined in the extreme position into which the vessel rolls, is 
specially worthy of observation. A strip of wood was fastened at right 
angles to that extremity of the yard to which the deflecting weight was 
attached, of sufficient length to dip into the water when the vessel rolled; 
on this slip of wood, and also on the side of the vessel nearest to it. a 
strip of glazed paper was fixed. The highest points at which these strips 
of paper were wetted in the rolling of the vessel, were obviously points 
in the water-line in its extreme position, and being plotted upon the board, 
a line drawn through them determined that position with a degree of 
accuracy which left nothing to be desired. 

1 wo tonus of vessels were used; one of them had a triangular and the 
other a semicircular section. The following table contains the general 
results of the experiments. 



ON THE KOLLING OF SHIPS. 



641 














, 


a -i 


8 J 


Into 
roll, 
Hon 


= -■- 


g |*g ! 










\l 


11 


•a 


si « 

Jig 1 


S = 

S >■ -- * 




Form of the 
model ex- 
peri meuted 
on. 


No. of 
experi- 
ment. 


Wei cut of 

mid. 1 and 
loading. 


Disturb- 
ing 
■weight. 


If 


*** 


I|| 


3"* x 


i - * „ li 










a 


iiii 

2g<e 


i 2 


Hi* J! 

- - - s » 

H]§5 


= .= c * 


Hii! 






ibs. 


lbs. 






o , 


. , 


o ■ 




Triangular 
model. 


1. 


83-8626 


•5485 


•5161 


•5361 


23 30 




12 30 


•8961 


2. 


36-8590 


•3450 


•4887 


•4951 


15 30 




8 


•98114 


3. 


37-3563 


•5377 


1-1724 


1-4503 


24 




13 


■88512 




4. 


38 2911 


•5739 


1-2673 


1-8460 


25 




13 30 


•9330 


Circular 
model. 


1. 


197-18 


2-8225 


7-8761 


7-394 


26 


24 20 


13 




2 


19718 


1-9570 


3-2486 


3-122 


17 


16 22 


9 




3. 


255-43 


1-9570 


1-7727 


1-7667 


10 


10 


4 30 





In the experiments with the smaller triangular model the differences 
between the results and those given by the formula are much greater than 
in the experiments with the heavier cylindrical vessel. 

In explanation of this difference, it will be observed, first, that the con- 
ditions of the experiment with the cylindrical model more nearly approach 
to those which are assumed in the formula than those with the other; the 
disturbance of the water in the change of the position of the former being 
less, and therefore the work expended upon the inertia of the water, of 
which the formula takes no account, less in the one case than the other ; 
and, secondly, that the weight of the model being greater, this inertia 
bears a less proportion to the amount of work required for inclining it 
than in the other case. 

The effect of this inertia adding itself to the buoyancy of the fluid, 
cannot but be to lift the vessel out of the water and to cause the displace- 
ment to be less at the termination of each rolling oscillation than at its 
commencement* This variation in volume of the displacement was appa- 
rent in all the experiments. Its amount was measured and is recorded 
in the last column of the Table; its tendency is to produce in the body 
vertical oscillations, which are so far independent of its rolling motion 
that they will not probably synchronise with it. The body, displacing, 
when rolling, less fluid than it would at rest, the effect of the weight 
used in the experiments to incline it is thereby inoreased, and thus is 
explained the fact (apparent in the eighth and ninth columns of the Table) 
that the inclination by experiment is somewhat greater than the formula 
would make it. 

The dynamical stability of a vessel whose athwart section* (where they 



* This result connects itself with the well-known fact of the rise of I 
out of the water when propelled rapidly, which i of fait 

track-boats, as considerably to reduce the resistance upon them* 

-n 



:x. 

are subject to immersion and emersion) are circular, having their centre* in 

a common axis. 

rig. i. rig. a. 





Let EDF, fig. 1. or 2., be an athwart section of such a vessel the 
parts of whose periphery ES and FR, subject to immersion and emersion, 
are parts of the same circular arc ETF. whose centre is C. Let G, repre- 
sent the projection of the centre of gravity of the vessel on this section, 
and G, that of the centre of gravity of the space whose section is SDRT, 
supposing it filled with water. The space lies wholly within the vessel in 
fig. 1. and without it in fig. 2. Let 

h i = CG,, h t = CG,. 

W, = weight of vessel. 

TV, = weight of water occupying, or which would occupy, the space 
whose section is STRD. 

= the inclination from the vertical. 

Since in the act of the inclination of the vessel the whole volume of 
the displaced fluid remains constant, and also that volume of which STRD 
is the section,* it follows that the volume of that portion of which the 
circular area PSRQ is the section remains also constant, and that the 
water-line PQ, which is the chord of that area, remains at the same dis- 
tance from C, so that the point C neither ascends nor descends. Now the 
a which constituted the equilibrium of the vessel in its vertical posi- 
tion were its weight and that of the fluid it displaced. Since the point C 
is not vertically displaced, the work of the former force, as the body 
inclines through the angle *, is represented by — W, fi, vers. 9. The work 
of the latter is equal to that of the upward pressure of the water which 
would occupy the space of which the circular area PTQ is the section 
■''.7. in the case represented in fig. 1.. by that of the water which 
would occupy STRD ; and diminished by it in the case represented in 

But since the space, of which the circular area PTQ is the section, 

remains similar and equal to itself, its centre of gravity remains always 

at the same distance from the centre C. and therefore neither ascends 

Whence it follows that the work of the water which 

a that the work of the whole displaced 

fluid is equal to that of the part of it which occupies the space STRD, 

* It will be observed that the space STRD is supposed always to be undef 
water. 



ON THE ROLLING OF SHIPS. 6^3 

taken in the case represented in fig. 1 . with the positive, and in that re- 
presented in fig. 2. with the negative sign. It is represented, therefore, 
generally by the formnla ±W,A 2 vers.0. On the whole, therefore, the 
work Xu 2 of those forces, which in the vertical position of the body con- 
stituted its equilibrium, is represented by the formula — 
Xu 2 = — W,^, vers. 9 ± W 2 h 2 vers. 9. 
Kepresenting, therefore, the dynamical stability Sw, by U (0), we have by 
equation (2. p. 122.) 

U (?) = (W l h, + W 2 h 2 ) vers. 0, 
in which expression the sign ~-f is to be taken according as the circular 
area ATB lies wholly within the area ADB, as in fig. 1 , or partially with- 
out it, as in fig. 2. Other things being the same, the latter is therefore a 
more stable form than the other. 

13. The work of the upward pressure of the water upon the vessel 
represented in fig. 2. being a negative quantity, — W 2 ^ 2 vers. 0, it follows 
that the point of application of the pressure must be moved in a direction 
opposite to that in which the pressure acts ; but the pressure acts upwards, 
therefore its point of application, i. e. the centre of gravity of the displaced 
fluid, descends. This property may be considered to distinguish mechani- 
cally the class of vessels whose type is fig. 1., from that class whose type is 
fig. 2. ; as the property of including wholly or only partly, within the area 
of any of their athwart sections, the corresponding circular area ETF, dis- 
tinguishes them geometrically. 

The dynamical stability of a vessel of any given form subjected to a roll- 
ing or pitching motion. 

Conceive the vessel, after having completed an oscillation in any given 
direction — being then about to return towards its vertical position — to 
be for an instant at rest, and let RS represent the 
intersection of its plane of flotation then, and PQ 
of its flotation when in its vertical position, with 
a section CAD of the vessel perpendicular to the 
mutual intersection O of these planes. The sec- 
tion CAD will then be a vertical section of the 




Let G be the projection upon it of the vessel's 
centre of gravity when in its vertical position. 

H, that of the centre of gravity of the fluid displaced by the vessel in the 
vertical position. 

g, that of the fluid displaced by the portion of the vessel of whir 

is a section. 
A, that of the fluid which would he displaced by the portion, of which 

POR is a section, if it were immersed. 
GM, HN", gm, hn, EL, perpendiculars upon the piano i: 
W = weight of vessel or of displaced fluid. 

w = weight of water displaced by either of the equal potions o( the 
vessel of which POR and QOS are Beotione. 



C>[t APPENDIX. 

II, = depth of centre of gravity of vessel in vertical position. 
II, = depth of centre of gravity of displaced water in vertical 
position. 
aH, = elevation of centre of gravity of vessel. 
&H, = elevation of centre of gravity of displaced water. 
P = area of plane PQ. 
6 = inclination of planes PQ and RS. 
rj = inclination of line in which planes PQ and RS intersect, 

to that line about which the plane PQ is symmetrical. 
h = perpendicular distance of line from centre of gravity of 

plane PQ. 
£ = inclination to horizon of line about which the plane PQ is 

symmetrical. 
x = distance of section CAD, measured along the line whose 
projection is 0, from the point where that line intersects 
the midship section. 

y = 0/3. 

y. = PQ. 

y. = RS. 

z = hn + mg. 

*. = KL. 

I = moment of inertia of plane PQ about axis 0. 
A and B = moments of inertia of PQ about its principal axes. 

H = weight of a cubic unit of water. 
Suppose the water actually displaced by the vessel to be, on the contrary 
contained by it; and conceive that which occupies the space QOS to pass* 
into the space POR, the whole becoming solid. Let aH 3 represent the 
corresponding elevation of the centre of gravity of the whole contained 
fluid. Then will aH 2 + aH 3 represent the total elevation of the centre of 
gravity of this fluid as it passes from the position it occupied when the 
vessel was vertical into the position PAQ. But this elevation is obviously 
the same as though the fluid had assumed the solid state in the vertical 
position of the body, and the latter had revolved with it, in that state, into 
its present position. It is therefore represented by KH — NH ; 

.-. aH 2 + aH 3 = KH — NH and aH 3 = KH — NH — aH 2 . 

Since, moreover, by the elevation of the fluid in QOS, whose weight is w, 
into the space OPR, and of its centre of gravity through (gm + hrt\ the 
centre- of gravity of mass of fluid of which it forms a part, and whose weight 
is W, is raised through the space AH 3 ; it follows, by a well-known property 
of the centre of gravity of a system,* that 

* The line joining the centres of gravity of the vessel and its immersed part, 
in its vertical position, is parallel to the plane CAD, for it is perpendicular tcr 
the plane PQ, to whose intersection with the plane RS the plane CAD is per 
peudicular ; . ■ . GK = H, and HE = H*. 



But 
and 



ON THE ROLLING OF SHIPS. 645 

"W. A H 3 = w (gm + hri) ; 
/. W(KH -M-aH 2 ) = m^ + h?i). 

1ZB. = KH cos. 8 — KL = H 2 cos. 9 — % ; 
■.KH— NH= Havers. 0+jt, 



m<7 + nA = s ; 
.*. W (H 2 vers. 9 + % — A H 2 ) = wz ; 

/. W. A H 2 = W (H 2 vers. 9 +%) — wz (28). 

Also A H x = KG — MG = H t — (H, cos. $ — %) = H, vers. 9+\; 

.'. W (A H, — A H 2 ) =: W (H, — H 2 ) vers, + ws; 
.-. (equation 27.) U (0, j?) = W (H, — H a ) vers. + wz ; . . . (29). 
If a/3 be a vertical prismatic element of the space QOS, whose base 
is dx dy cos. 0, and height y sin. then will w .mg be represented, in 

respect to that element, by ny sin. 9. dx dy cos. 9.%y sin. 0, or by ~ ,usin. a 

cos. y*dx dy ; and wz will be represented, in respect to the whole space 
of which PrsQ is the section, by 

o fi sin. 8 6 cos. 6 j fy^dx dy, 

or by o jt sin. 3 cos. 0. 1. 

If therefore we represent by $ the value of wz, in respect to the spaces 
of which the mixtilinear areas PRr and QSs are the sections, we have 

1 

wz = o ;" I sin. 2 cos. + $. 

But the axis 0, about which the moment of inertia of the plane PQ is 

I, is inclined to the principal axes of that plane at the angles rj and„ — ^, 

about which principal axes the moments of inertia are A and B, 
.-. I = A cos. 2 v + B sin. 2 n + Ph\ 
.•.U(0,,)= 

W (H, — H 2 ) vers. 6 + -p (A cos. 2 n + B sin. 2 n + PA 2 ) sin 2 9 cos. + $... (30). 

It has been shown by M. Dupin* that when Q is small the line in 

* Surla Stability des Corps Flottants, p. 32. In calculations having refer- 
ence to the stability of ships, it is not allowable to consider 6 extremely small, 
except in so far as they have reference to the form of the ship immediately 
about the load-water line. The rolling of the ship often extemls to 20° or 80°, 
and is therefore largely influenced by the form of the vessel beyond these 
limits. Generally, therefore, equation 30. is to be taken ai that applicable to 
the rolling of ships, those which follow being approximations only applicable 
to small oscillations, and not sufficiently near (excepting equation W) foi 
practical purposes. 



(UO 



APPENDIX. 



which the planes PQ or RS intersect passes through the centre of gravity 

of each ; in this oase 

.-. I = A cos. 1 r { + B sin. 2 ^ ; 

therefore by equation (30), 

U (0^) = W (II , — H,) vers. + -■ ^ (A cos.* n + B sin.' n ) sid.* cos. + $>. 

If be so small that the spaces P/-R and Q*S are evanescent in compari- 
son with POr and QOs, then, assuming $ = and cos. 0=1, 

U (0, jy) = W (II, — H 2 ) vers. 8 + - p (A cos.' »? + B sin.* n ) sin.* 6 

which may be put under the form 

U (0, n ) = | W (H» — H 2 ) + /* (A cos.' ^ + B sin. 2 ,) i vers. 0. 



(31), 



in, since 

sin. £ = sin. sin. >?,.... (32), 

and (Acos.'j; + B sin. 2 ??) sin. 2 6 = {A -f (B — A) sin. 2 *;} sin. 2 0, 

.-. (A cos.^ + B sin'r;) sin. 2 = Asin.*0 + (B - A) sin. a £; 

/.by equation 31, 

U(0,O = W(H,— H 2 )vers.0 + 2 >{Asin. 2 + (B — A)sin. 2 ^}, (33), 

by which formula the dynamical stability of the ship w represented, both 
as it regards a pitching and a rolling motion. 

If in equation 31. 17 = -, the line in which the plane PQ (parallel to the 
2 

deck of the ship) intersects its plane of flotation is at right angles to the 

length of the ship, and we have, since in this case = £ (see equation 32.), 

U(Q = W(H } — H 2 ) vers£+ |^Bsin 2 £ (34), 

which expression represents the dynamical stability, in regard to a pitch- 
ing motion alone, as the equation 

U(0) = TV(H,— H 2 ) vers 0+ | M Asin 2 (35), 

represents it in regard to a rolling motion alone. 

16. If a given quantity of work represented by U(0) be supposed to 
be done upon the vessel, the angle through which it is thus made to 

roll may be determined by solving equation 35. with respect to sin.-. 

2 

"We thus obtain 

. .0 W(H,— HQ+jiA— VTW(H,— H 2 ) +A tA} 2 — 2^A.U(0 ) , . . (3G ^ 
8in * 2~ 2^*A 






ON THE ROLLING OF SHIPS. 647 

17. If PR and QS be conceived to be straight lines, so that POR and 
QOS are triangles, then w. z, taken in respect to an element included 
between the section CAD, and another parallel to it and distant by the 
small space dx, is represented by 

— WiVi sm.edx(mg + iih) ; 
4 

or, since mg+nh=-y 1 sin.0, 

3 

by _ (4, smSdytyzdx ; 

12 

. • . wz = — fi sin. 2 / y\y 2 dx, 
12 J 

and} equation 29 

U(0,O=W(H 1 — H 2 ) vers.0 + 1 psin. 8 * fy\y£x, . . . (37), 
24 «/ 

which formula may be considered an approximate measure of the stability 

of the vessel under all circumstances. 

If, as in the case of the experiments of Messrs. Fincham and Rawson, 

the vessel be prismatic and the direction of the disturbance perpendicular 

to its axis, 

j/ x = constant = #, and z = —a sin.0 ; 

o 

,-.wz = —aw sin. and 
3 

U(0)=W(H t — HOvers.0+iow sinA 
3 

A rigid surface on which the vessel may oe supposed to rest whilst in the 
act of rolling. 

If we imagine the position of the centre of gravity of a vessel afloat 
to be continually changed by altering the positions of some of its con- 
tained weights without altering the weight of the whole, so as to cause 
the vessel to incline into an infinite number of different positions dis- 
placing, in each, the same volume of water, then will the different plane; 
of flotation, corresponding to these different positions, envelope B oarve I 
surface, called the surface of the planes of flotation {svrfu-e dajlotabona), 
whose properties have been discussed at length by M. Dupis In hifl 62 
cellent memoir, Sur la Stabilite des Corps Flottants, which tonus part of 
his Applications de Greometrie* So far as the properties of this Burfeoe 
concern the conditions of the vessel's equilibrium, tlie\ have been 62 
hausted in that memoir, but the following property, which has Nfemot 



* Bauhslieu, Paris, L822. 



648 APPENDIX. 

rather to the conditions of its dynamical stability than its equilibrium, ia 
not stated i». M. Di'i-ix : — 

If ire conceive the surface of the jtlanes of flotation to become a rigid 

>' also the surface of the fluid to become a rigid plane without 

friction, so that the former surface may rest upon the latter and roll and 

i>on it, the other parts of the vessel being imagined to he so far im- 

■ il as not to interfere with this motion, but not so as to take away 

their tceight or to interfere with the application of the upward j> restore of 

the fluid to them, then will the motion of the vessel, when resting by this 

curved surface upon this rigid but perfectly smooth horizontal plane, be 

the same as it was when, acted upon by the same force, it rolled and pitched 

in the fluid. 

In this general case of the motion of a body resting by a curved sur- 
face upon a horizontal plane, that motion may be, and generally will be, 
of a complicated character, including a sliding motion upon the plane, 
and simultaneous motions round two axes passing through the point of 
contact of the surface with the planes and corresponding with the rolling 
and pitching motion of a ship. It being however possible to determine 
these motions by the known laws of dynamics, when the form of the 
surface of the planes of flotation is known, the complete solution of the 
question is involved in the determination of the latter surface. 

The following property*, proved by M. Dupin in the memoir before 
referred to (p. 32), effects this determination : — 

" The intersection of any two planes of flotation, infinitely near to each 
other, passes through the centre of gravity of the area intercepted upon 
either of these planes by the external surface of the vessel." 

If, therefore, any plane of flotation be taken, and the centre of gravity 
of the area here spoken of be determined with reference to that plane of 
flotation, then that point will be one in the curved surface in question, 
called the surface of the planes of flotation, and by this means any number 
of such points may be found and the surface determined. 

The axis about which a -vessel rolls may be determined, the direction in 
which it is rolling being giren. 

If, after the vessel has been inclined through any angle, it be left to 
itself the only forces acting upon it (the inertia of the fluid being neglected) 
will be its weight and the upward pressure of the fluid it displaces ; the 
motion of its centre of gravity will therefore, by a well-known principle 
of mechanics, be wholly in the same vertical line. 

Let UK represent tins vertical line, PQ the surface of the fluid, and 
aW> the surface of the planes of flotation. As the centre of gravity G 
traverses the vertical ELK, this surface will partly roll and partly slide 
by its point of contact M on the plane PQ. 

If we suppose, therefore, PPwQ to be a section of the vessel through 

* This property appears to have been first given by Eulkr. 






OX THE EOLLIXG- OF SHIPS. 



619 



Fig. 8. 




the point M, and perpendicular to the axis about which it is rolling, and 
if we draw a vertical line MO through the point M, and through G a 
horizontal line GO parallel to the plane PRQ, then 
the position of the axis will be determined by a line 
perpendicular to these, whose projection on the plane 
PEQ is O. 

For since the motion of the point G is in the verti- 
cal line HK, the axis about which the body is revolv- 
ing passes through GO, which is perpendicular to 
HK ; and since the point M of the vessel traverses 
the line PQ, the axis passes also through MO, which 
is perpendicular to PQ ; and GO is drawn parallel to, and MO in the 
plane PRQ, which, by supposition, is perpendicular to the axis, therefore 
the axis is perpendicular to GO and MO. 

If HK be in the plane PRQ, which is the case whenever the motion is 
exclusively one of rolling or one of pitching, the point is determined by 
the intersection of GO and MO. 

The time of the rolling through a small angle of a vessel whose athwart 
sections are (in respect to the parts subject to immersion and emersion) 
circular, and have their centres in the same longitudinal axis. 

Let EDF (fig. 1. or fig. 2Jl represent the midship section of such a 



Fig. 1. 





vessel, in which section let the centre of gravity G, be supposed to be situ- 
ated, and let HK be the vertical line traversed by G, as the vessel rolls. 
Imagine it to have been inclined from its vertical position through a given 
angle 6 l and the forces which so inclined it then to have ceased to act 
upon it, so as to have allowed it to roll freely back again towards its posi- 
tion of equilibrium until it had attained the inclination OCD to the verti- 
cal, which suppose to be represented by 0. 

Referring to equation 1. page 122. let it be observed that in this OBM 
2^=0, so that the motion is determined by the condition 

Xu^^-^wv 2 (38). 



But the forces which have displaced it from the position in which it 
as, for an instant, at rest are its weight and the upward pressure of the 



G50 ; J.M-IX. 

: uid the work of th< — U(0), done between the inclination* 

i when the vessel was in the sot of receding from the vertical, was 
shown to be represented by (WATWaJi*) (vers. 6 — vers. 0,); therefore 
the work, between the same inclinations, when the motion is in the 
opposite direction, is represented by the same expression with the sign 
changed j 

.-. Eii 1 =(W 1 * 1 TW J *0(verB. 0, — vers. 0), 

and since (he axis about which the vessel is revolving is perpendicular to 
the plane EDF, and passes through the point O, if W^-* represents its 
moment of inertia about an axis perpendicular to the plane EDF, and 
passing through its centre of gravity G„ 

Substituting in equation 38. and writing for OGi its value A, sin. 0, we 
have 

(W.A.TVA) (vers. 0, — vers. *)= ?J Q*+h* sin.'e) (^) ; 
•'* W = , —1 w 7, C /*■+*? sin.«e „ 



i?+4»!*m.'j 5 #ooa' s 




V^^Wx) / V o^.l^in,^) 



<Zfl = 



V*( 1:f wa) 




Fsec'^e+^sin.^e , 
j j_, cos. - «Z T a, 

sin.* 2^1— sin.^ 9 



-lining to be so small that the fourth and all higher powers of 
si n. — 6 may be neglected, and observing that, this being the case, 

^JV sec* \ e+ 4A?sin.'i e = * /Wl+sin.' \ 9 \+4*5 sin.* \ e 



OX THE EOLUNG OF SHIPS. C51 



+0, 

1 ~ f ~2F~ Sm - V 1 

V-{,-.i.4"' 5 



+ 0, 



/ 



,7 • * 

a sm.— 

a 



V 



and 



+ 0i 



/ 



v/ 



sin. ! i^sin.l 9 } j 

. ,1 . ,1 = 2" 3in '2'» 

sin. 2 - 0! — sin. -0 



The sign + being taken according as the centre of gravity of the displaced 
fluid ascends or descends. 

The time of a vesseVs rolling or pitching through a small angle, its form 
and dimensions oeing any whatever. 

Let EDF (figs. 1. or 2.) represent the midship section of such a vessel, 
supposed to be rolling about an axis whose projection is ; and let C 
represent the centre of the circle of curvature of the surface of its planes 
of flotation at the point M where that surface is touched by the plane PQ, 
being above the load water-line AB in fig. 1, and beneath it in fig. 2. Let 
the radius of curvature CM be represented by p; then adopting the same 
notation as in the last article, and observing that the axis O about which 
the vessel is turning is perpendicular to EDF, we shall find its uioiik 
inertia to be represented by 

W 1 {F+(H 1 -p)«aiii.^j^ 

where H, represents the depth of the centre of gravity in the vertical posi- 
tion of the vessel. 



0.)^ AIM'i \I>IX. 

Also, by < .-nation 35. 
-W|=U(8|) — U(0) = W,(II 1 -II,)(C()S.d — COS. 0.) +1^(003.'©— cos.'fl,), 

,•. by equation 38. 

^ l aI|-^ 2 )(cos.O-cos.o,) + |^A(c(»s. :l O-cos. s O J )=|- , I ^ + (H 1 -p) a sin.»0 | (*Y 



w= 



w= 




F + (H,-p) 3 sin. a e 



l^A, 



(H.-H,) (cos. 0-cos. 0.) +- —(cos. 9 — CO8. 9 0,) 



^ + (H I - P )'8in.'tf 



■J H,-H 2 + -'^-(cos. 0+cos. 0.) U cos. o — cos. 0, i 



Assuming and 0, to be so small that cos. + cos. 9. = 2, and observing 
that 



cos. — cos. X = vers. 0, — vers. 0, 



-0i 
Supposing, moreover, P to remain constant between the limits— 0, and -f-0„ 
and integrating as in equation 39. 



*(»i) = 



*& 



y^n.-H,- 



£A 



1+ w sin.'-0. 



. . (41). 



Since the value of sin.^0, is exceedingly small, the oscillations are 

nearly tautochronous, and the period of each is nearly represented by the 
formula 



1(9,) = 



itk 



\A(h,- h ' + wT) 



Ws'--- m 



EQUILIBRIUM OF PRESSURES. 653 

The following method is given by M. Dupist for determining the value 
of P *:— 

" If the periphery of the plane of flotation be imagined to be loaded at 
every point with a weight represented by the tangent of the inclination of 
the sides of the vessel at that point to the vertical, then will the moments 
of inertia of that curve, so loaded, about its two principal axes, when 
divided by the area of the plane of flotation, represent the radii of greatest 
and least curvature of the envelope of the planes of flotation." 

If p be taken to represent the radius of greatest curvature, the formula 
41. will represent the time of the vessel's rolling; if the radius of least 
curvature (B being also substituted for A), it will represent the time of 
pitching. 

NOTE D. 

On the conditions of the equilibrium of any number of pressures in the 
same plane, applied to a body moveable about a cylindrical axis in the state 
bordering upon motion. (From a memoir on the Theory of Mechanics, 
printed in the second part of the Transactions of the Royal Society for 1841.) 

Let Pi, P 2 , P 3 , &c. represent these pressures, and R their resultant. Also 
let «!, a 2 , a 3 , represent the perpendiculars let fall upon them severally from 
the centre of the axis, those perpendiculars being taken with the positive 
signs whose corresponding pressures tend to turn the system in the same 
direction as the pressure P„ and those negatively which tend to turn it in 
the opposite direction. Also let -k represent the perpendicular distance of 
the direction of the resultant R from the centre of the axis, then, since R 
is equal and opposite to the resistance of the axis, and that this resistance 
and the pressures P w P 2 , P 3 , &c. are pressures in equilibrium, we have by 
the principle of the equality of moments, 

Pi«! -f P 2 # 2 -f- F 3 a 3 + &c. = tJR. 
Representing, therefore, the inclinations of the directions of the pressures 
P„ Pa, P 3 , &c. to one another by *,.„ «, 3 , « 23 , t, &c, &c, and substituting 
for the value of R.J 



* Applications de Geometrie, p. 47. 

f The inclination »,., of the directions of any two pressures in the above ex- 
pression is taken on the supposition that both the pressures act from, or both 
towards the point in which they intersect, and not one toward*, and the other 
from, that point; so that in the case represented in the figure in the note ut p, 
175., the inclination c I2 of the pressures P, and 1\„ represented by the arrows, 
is not the angle P, IP 2 , but the angle P,IQ, sinoe [Q and IP, are directions of 
these pressures, both tending //vm/ this point of intersection, whilst, the direc- 
tions of P 2 I and IP, are one of them towards that point, and the other from it. 

\ Poisso.v, Mecanique, Art. 33. 



!',M 

PjOi-t-PtOfeH iv. • 



Pi= 

P a O a + P»ga + . 

«1 



APPENDIX. 



P, , +P, , +P. , .+ ... 

■f2P,P,co8. v. + 2P,P 3 cos. « I3 + . . 
+ 2 P,P, cos. < 2 . 3 + 2 P 2 P 4 cos. i w + . . 
■f&C. &C. 



J 



{P,' + 2P l (P,cos. 1,., + P.cos. i,, + . . . .) 
+ p/ + p,« + p 4 » + ... 
+ 2 P 2 P 3 cos. « 5 .3+2P 2 P 4 cos. « a . 4 
+ &c. &c. 



If the value of P, involved in this equation be expanded by Lagrange's 
theorem * in a series ascending by powers of a,, and terms involving powers 
above the first be omitted, we shall obtain the following value of that 
quantity : — 



p __ P 2 ff 2 + P 3 a 3 +. 



G) 



or reducing, 



T (P 2 tf 2 + P 3 a 3 + P 4 a 4 + )' 



-(T 2 a t + Y 3 a 3 + 'P A a i + , 
a. 



(P a COS. «,« + P 8 COS. «,.j + P 4 COS.i w + . . . .) 

+ P a 2 +P- ! 3+P 4 3 + .... 

+ 2 P 2 P 3 cos. i 23 + 2 P 2 P 4 cos. «2. 4 
+ 2P 3 P 4 cos.« 34 + 



_ P 2 ^ + P 3 ^3 + ... 



P«*( a i* — 2a,02cos. i rf + OJ 8 ) 
+ P 3 2 (a, 2 — 2a,a 3 cos. i M + a 3 *) 
+ &c. <fec. 

+ 2 P 2 Ps{a-A — «i(flh cos.t 2 .j 4- a 2 cos. t M + a 3 cos. i„) j 
+ 2 P, P 4 {a a a 4 — a, (a, cos.t 2 . 4 -f a 2 cos ,„ 4- a x cos. »,.,) ! 
+ &c. &c. 



Now a' — 20,fl 2 «cos. «, 2 + a 2 represents the square of the line joining the 
feet of the perpendiculars a, and # 2 let fall from the centre of the axis 
upon P, and P 2 ; similarly a\ — 2a t a 3 cos. i,. s + a| represents the square of 
the line joining the feet of the perpendiculars let fall upon P, and P 3 , and 



' I'll is expansion may be effected by squaring both sides of the equation, 
solving the quadratic in respect to P„ neglecting powers of X above the first 
and reducing ; this method, however, is exceedingly laborious. 



ROLLING MOTION OF A CYLINDER. 655 

ro of the rest. Let these lines be represented by L 12 , L 2 3 , L, 4 , &c, and let 
the different values of the function 

{a 2 a 3 — a x {a v cos. c 2 . 3 + a^ cos. i Vi + a 3 cos. c^)} 

be represented by M 2 . 3 , M 2 . 4 , M 34 , &c. 



.•.P 1= 



P, ** + P 3 ^3 + . .. , Ji (P 2 2 L,. 2 2 + P 3 2 L, 3 2 + P 4 2 L, 4 2 +....)£ 

a, + o~A + 2P 2 P 3 M 23 + 2P 2 P 4 M 24 + . . . J • " <** 



NOTE E. 

On the Polling Motion of a Cylindee. 

(From a memoir printed in the Transactions of the Royal Society for 
1851, part II) 

The oscillatory motion of a heterogeneous cylinder rolling on a horizontal 
plane has been investigated by Euler.* He has determined the pressure 
of the cylinder on the plane at any period of the oscillation, and the time 
of completing an oscillation when the arcs of oscillation are small. 

The forms under which the cylinder enters into the composition of 
machinery are so various, and its uses so important, that I have thought it 
desirable to extend this inquiry, and in the following paper I have sought 
to include in the discussion the case of the continuous rolling of the cylin- 
der, and to determine — 

1st. The time occupied by a heterogeneous cylinder in rolling continu- 
ously through any given space. 

2ndly. The time occupied in its oscillation through any given arc. 

3rdly. Its pressure, when thus rolling continuously, on the horizontal 
plane on which it rolls. 

Under the second and third heads this discussion has a practical appli- 
cation to the theory of the pendulum ; determining the time occupied in 
the oscillations of a pendulum through any given arc, whether it rests 
on a cylindrical axis or on knife-edges, and the oiroumstances trader 
which it will jump or slip on its bearings; and under the fust and third, 
to the stability and the lateral oscillations of locomotive engines in rapid 
motion, whose driving-wheels are, by reason of their cranked axles, untruly 
balanced. 



* Nova Acta Acad. Petropol. 1788. " De motu osoillatorio oirCfl 1X00 eylin- 
dricum piano horizontali incumbentem." 






APPENDIX. 




taon of a I 

G and perpendicular to : 

time, with the hori- 
zontal plane BD on which it is rolling. 
Assume 
a = AC.h = CG.r = j, 
"W = weight o: ^Vi J = momen- 

tum of inertia of the cylinder about 
an axis passing through G and 
parallel to the axis of the cylinder. 

u = gj ren value of the angular velocity ( — ) when has the _ 
value 0,. 
= given value of * when the angular velocity h* 
I = given value of GM corresponding to the value 0, of 6. 

Then W = W< I- - a 1 - h* — Soft cos. e) — moment of inertia 

about M. Since moreover the cylinder may be considered to be in the act 
of revolving about the point M by which it is in contact with the plane, 
one-half of its tis rtV the formula 



If 

- 



. 



U)' 



and one-half of the rw rtra acquired by it in rolling through the angle 



• - 



j l>-cr-2ahcos.i-h^) -<*+* 

But the vertical descent of the centre of gravity while the cylinder is 
passir.j n into the other, is represented by 

h (cos. 6 — cos. B t ). 
inciple of rt* r 



l-i^-ua 1 — 2«i cos.d+aV-)— (¥-'■ :os. f - cos. O, 

jtain 

~ C03.fl 1 ) + (P-|-7 I )u > 

\dt) ~ - f — 2ah cos. e+ tf 



whence we obtain 



m % COS. 9 — (cOS. 0, — — S, 

(9\ V tyh J 

-\aJ 1/F a h\ 



.a). 



• Robbob, Dynmmiqwt, 2** partie, 565.; Poxcelet, Mecaniqtu Induttritlk, 
orArt 



ROLLING MOTION OF A CYLINDER. 65' 



/3 = cos.0 r 



<TSf) - (8) - 



,.; * /f\*/~! = i\» sod «=_n* f(£Z*SJ$ 

dt \a) \a — cos. 6 J ' W,/ V cos - y — 3 ' 



■•-©*/(=S)**-^ 



where £ represents the time of the body's passing from the inclination X to 
zero. 

Now let it be observed that in this function a>3 so long as a is less than 
g, since 

& + }*>— (tf + P)J>, or ¥ + 0?— 2ah cos. fl, + A 2 > — (# + F)»» 
and . • . F + a 2 + h*> 2ah cos. 6 t — (*■ + Z^*, 

1/F a A* fl F + Z 2 2 

and 2U' + A + a) >COS ' 9 '-^rA' W - 

1+a 1 — a , a — CO'*. fl 

Let 1^=^ -r=5 =? ' ^aT?=F = ^ seo ^' 

Then when = 0, a 2 sec. 2 ? = £ ; = ?', • ' • sec. </. = 1 and ^ = 0. 

1 — ,J 

When = 0i let ^ = </»„ 

1 / F h a\ 

lijh + a + l)-^^ 

a — cos. 0, x ' 



•.^sec. 2 ^=- ^ 

2^; 



cos. ^-0 r^v 



,ff v F + a 2 + fr 2 — 2aft cos, fl, 9_ % 

~\a) (#*+£')«' — aw 2 ' 

also 

1/7j ? a h\ 



1-a 



j. / a, a /6\ 

"2\^ + ^ + flj - g + ( a — fry 



! - "iZji - "" " . /F + Pv t U hgh vers. 0, + (F TTV' 

(£ 2 +ZV + 2<7^vers. fl, 
.•.sec. 2 ^= { # + (a _ A)> . i • • • • ( 5 )- 

Now / (S=l=if) dJ= f ( c^^-T) <*/'* 

o 

. . . a — cos. 

And since = Q sec/ <£, 

cos. 0-/3 * 

2 cos. A — (a + |3) _ cos. 2 <p — q* 
(a — J3) - co<. 2 </, + 7" 

(a + 0)(cos. 2 <p + y ? ) + (a — 3) (cos : V = Y3 
. • . 2 cos. 0= (cTis.^ + r) 

42 






ArPEXPIX. 



o I - 

008 6=— 6). 



(COS. 1 . + f7 

_ (co>.' f + g» — o cos.* f — -j-g'-oeos.': 

(cos. , .+_.» 
= {(1 — 3)?* + (l— o)c05.*f}{(l+.3>f + (l+a)cos.»d} 
(? S +C03.',)» 



Now 



* 



- ' 



... (6). 



ML <* * 4 COS. p * 4f * j co». p 

Also by equation (6.), 

_ 2a(y , -fcos. , » c^. :-i _c m ; - :; : mm _. a — 3_*c<w.» 

ioo&t _ , -hcos. , .) , ~~ -." : 77~ ; 
. • . by equations (7.) and - 

tfj [_- (* — 3 )g* g ? + cos.'f c -. f 

: — ; : : ' 7 L ~- ~ "-■ •- ■ " v- -.»./ 

2 a— COS.* 



/ a — C >*. Nt ^_'2 a — 3 7 * I 1 ) 

m ''\m.9—fi) dfT {1— r^-cos.'^-^cos.'.)* f 



2 a — 3 _ ' f 



1 



2(o— _ 1 



r-r 



if 



_____ --■- s ! 

.(9). 






1 



:-. : 



1— 



1 _______ 

1 — a a — 3 
1^3 



and 



,_ t 



1 -3 



_P* + _^ 1 +o 1-— I 



(l-j-g)(l — 3) 

- 



(10). 



ROLLING MOTION OF A CYLINDER. 650 

//a — COS. 6y<l9 

__ 2' g-% ! /• d4 

(1 — F)*{p* + qy(l + q*) J (1 — n sin. 2 <p)(l — c l si n. a #* 

2( a — % 2 / v 

■where n( — tmja,) is that elliptic function of the third order whose par* 
meter is — n and modulus c. 

NOW -r--^= y- - r- = t /jEZ 

■ + ?•)* (/l+a\ ,J-ax)» V"2S=5) 



j(&)-(£)} 



1—1 



i +r' . /i— • »\ U— ^/ 

2(a — /?Y 2(o — 1) 

• ( i_ (nw+sya+srvw 1 ^) 

/ V a h \ 



V(s + * + i)" 8 



(fc-'+JV 

COS.0,+ 






yJaTi(^+V)(l + ^ 



-,.... (12). 



. *.by equations 11. and 4. 

*=-7 F + ' a - A) ' . .n(-n.,),....(13). 

V^+'P + t) 

where (9.) (2.) (3.) 

1 — co3. 0,+ s~r» 2Avers.0, + — -m 1 

1— •* _ tyh _ a fr ,-. r, 

n= a — /?" 1/* « A\ " P + P , ,„ .., /, *A 

2(S + A + «)- t ' 0S -°' + -^-- "■>"(" .,) 



• I cannot find that this function has before been integrated, except in the 

ease in which i.s exceedingly small. 



1>0*0 APPENDIX. 

and (10.) (2.) (3.) 



1 > (-r + - + > )+ 1 \\ vers.0,+ . W [ah 

, = (i+a)(i— A^V* 4 " V >' 2 ? 4 > _ 

The value of n( — ncf> x ) beiftg determinable by known methods (Legex- 
dke, Fonctions Elliptiques, vol. i. chap, xxxiii.), the time of rolling is given 
by equation 13. 

In the case in which the rolling motion is not continuous but oscillatory, 

we have u = 0; and therefore (equation 5.) $, -*; n( — nctf>,) becomes 

Z 

therefore in this case a complete function. 

To express the value of this complete elliptic function of the third order 

in terms of functions of the first and second orders, let 



• o n 2 ±ah „„, 

sin. 2 «/, = -= =fr-F *\, (16)- 

& 1+a k+{a + hf 



Then' 



n(- ncl ) =F (4) +(T ^^{ F ( c |).E^- E (^).F,4 

Representing therefore the time of a semi-oscillation by £„ 

where (15.) c* = -^ik^+l') vers $l ^ 18 ^' 

Since the values of elliptic functions of the first and second orders, 
having given amplitudes and moduli, are given by the tables of Legendre, 
it follows that the value of t is given by this formula for all possible values 
of c and $. 

If the angle of oscillation 0, be very small e is very small, so that its 
square may be neglected in comparison with unity. In this case 

* Legendre, Calcus des Fonctions Elliptiques, vol. i. chap, xxiii. Art 116. 



ROLLING MOTION OF A CYLINDER. Q^J 



~Fc± = E^ = 4, and Fc * = Ec* = * 
2 2 2' 



.'.Fc^E^— Ec*Fc4, = 0. 
2 2 

For small oscillations therefore 

If the pendulum oscillate on knife-edges a=0,l = 7i, and we obtain I he 
well-known theorem of Legexdee (Fonctions Elliptiques, vol. i. chap, 
viii.) 



-v/^<1> •••(-)• 



where (18.) c* = \ vers, e, = sin. 2 ?!, 



. c = sin. * 0, (21). 

2 



In the case of the small oscillations of a pendulum resting on knife-edge, 
equation 20. becomes 



k / -- + — * (22) 

V gh - 2' K h 



which is the well-known formula applicable to that case. 
If the pendulum be one which for small arcs beats seconds (21.), 



/* + # 



2F 



'(•3 



. • . (20.) 2t = -— , (23). 



by which equation the time of the oscillation through any arc, of a pen- 
dulum which oscillates through a small arc in one Becond, may be deter- 
mined. I have caused the following table to be calculated from it. 



C(J2 



APPENDIX. 



Table of tlie time occupied in oscillating through every two degrees of a 
complete circle, by a pendulum which oscillates through a small arc in 
one second. 



Are ofo«cill:ition 


Time of one 


Arc ofoicillMion 


Time of one 


Arc nf o 


Time of one 


on well M'le »f 


complete 


on each "Me ui 


compleie 


■ n i ncli i>iile ol" 


Biiuipieie 


ill-- vertical 


OM'lll.lllUII III 


Die vertical 


oscillation in 


tlie vei lical 


osvillniion in 


In degrees. 


fecuii *. 


in degrees. 


second*. 


in degree*. 


second*. 


92 


1-1899 


122 


1-3905 


152 


1 -8032 


94 


1-20(10 


124 


1-4 o-Si) 


154 


1-8478 


96 


1-2123 


126 


1-4 2 S3 


156 


1-8963 


98 


1-2210 


128 


1 4486 


158 


1-9491 


100 


1-2322 


130 


1-4698 


160 


2-< 11)75 


102 


1 2439 


132 


1-4922 


162 


2-0724 


104 


1-2560 


134 


1-5157 


164 


2-1453 


100 


1-2686 


136 


1-5405 


166 


2-2285 


108 


1-2817 


138 


1-5667 


168 


2-8248 


110 


1-2953 


140 


1-5944 


170 


2-431)3 


112 


1-3099 


142 


1 -6238 


172 


2-58U1 


114 


1 -3249 


144 


1-6551 


174 


2-7621 


116 


1-3400 


146 


1-6884 


176 


3-0193 


118 


1 -3560 


148 


1-7240 


178 


3*4600 


120 


13729 


150 


1-7622 


180 


Infinite. 

I 



Tlie pressure of the cylinder on its point of contact with the plane on which 

it rolls. 
Let A' be the point where the point A of the cylinder was in contact 
with the plane. 

Let A'N = x, NG = y. 
\V — X = horizontal pressures on M in direction 

AM. 
Y = vertical pressure on M in direction MC. 
3 m m a' d Since the centre of gravity G moves as it would 

do if, the whole mass being collected there, all the impressed forces were 
applied to it, we have, by the principle of D'Alembert, 




g dt* ~ x 

9 dt 



.... (28). 



But since CA = a, CG = A, MCA = 0, 
. • . x — ad — h sin. 6, 
y — a — h cos. ; 



* Art. 96, equations (73.), (74.). 



ROLLING MOTION OF A CYLINDER. 

dd 



663 



Tt = hsin - d di 

d 2 x 



-d9 ' 






(29). 



by equation (29.), 



d 2 x 

7i~: 



t x = MA sin. — N(a — A cos. 0) 



d\j ,„ 

— _ = MA cos. — ]STA sin. y ; 



by equation (28.), 

x=^j_ Msin . e+N( «_ C0S . fl )n 

Y = W + jMcos.0 — Nsin.flV. 

But by equation (1.), substituting — and — 0, for and 0„ 



(30). 



»-©'- 



2gh(cos. e _ cos. 0.) + {¥ + PJo' 



& 2 + a 2 — 2ah cos. « + A 2 



. . . . (31). 



-© 



2aA(cos. 6 — cos. 0,) + (F + Z ; ) — 
{7 



A* + a 2 — 2aA cos. + A 2 



/ _x P + a 2 + A 2 - 2«A cos. 0, + (# + Z 2 )— — (V + a 2 + A' — 2aA co*. 9) 



& 2 + a 2 — 2aAcos. + A 2 



M 






2aAcos. ; 

Observing that a 2 + A 2 — 2ah cos. 0, = Z 2 . 

Differentiating this equation and dividing by( —J, 
[dtj (&' + a' + /t"-2aAcv 



(32). 



(38). 



AFFTCHX. 

Substituting these values of M and N in equation (30.), and reducing, 

W / ; »ir..-- - ( _ (P+PXP + &'— ah cos. ey g + **,/ )) 

a \ gQ? T a' + h' — 'lah cos. ey j { ' J * h 

■ytation of a body about a eyliudrieal axis of small diameter. 

:aing a = in equations (31.), (33.), and 9,= 0, we have 





0?i n 


fore, by equation (34 





X = _ - y + y — ,-:::• • . - - - (40). 

_ _ WAC?ft(3 oos*0— -Scos. 0—1) . ) 

1 = v ' - y\ — ft^ - ; • • • ( 41 >- 



The last equation may be placed under the form 



Y = W 









1/F+V \ 

If ~l . w 1 — I J be numerically less than unity, whether it be pos 

there will be some value of S between and * for which this 
will be equalled, with an opposite sign, by co*. 0, and for which 
the first term under the bracket in the value of Y will vani-h. This cor- 
responds to a minimum value o: :ted by the formula 

if zl -£-?-*•* — 1 I he numericallv greater than nnitv, then the 
3\ _ / 

minimum of Y will be attained when : = .t, and when 
s ( *+* I 



ROLLING Y^iION OF A CYLINDER. (jQl 



The Jump of an Axis. 

If Y be negative in any position of the body, the axis will obviously 
jump from its bearings, unless it be retained by some mechanical expe- 
dient not taken account of in this calculation. But if Y be negative in 
any position, it must be negative in that in which its value is a minimum. 
If a jump take place at all, therefore, it will take place when Y is a mini- 
mum ; and whether it will take place or not, is determined by finding 
whether the minimum value of Y is negative. If therefore the expression 
(42.) or (43.) be negative, the axis will jump in the corresponding case. 
An axis of infinitely small diameter, such as we have here supposed, 
becomes a fixed axis ; and the pressure upon a fixed axis, supposed to 
turn in cylindrical bearings without friction, is the same, whatever may 
be its diameter; equations (40.) and (41.) determine therefore that pres- 
sure, and equation (42.) or (43.) determines the vertical strain upon the 
collar when the tendency of the axis to jump from its bearings is the 
greatest. 



The Jump of a Rolling Cylinder. 

Whether a jump will or will not take place, has been shown to be deter- 
mined by finding whether the minimum value of Y be negative or not. 

Substituting a for -( _ + - + f \ and reducing, equation (35.) becomes 
Z\ah a hf 

or 

„ mt h \ W(^ + %r+Wjj ' a' — 1 { 

Y = W^l_cos.ej K — ^ ^-(a-cos.^' ■ ■ • W- 

W = ^\a 2^ (a — cos^T f co ~ tf+ 2gra*(tt— cos. )' 

• *I-0 1st when'-^^^ + ^r'^O, 2ndlv, when #-«, 
' 'Hi ^ ' a 2ga\ a — cos. ef 



3rdly, when 0=0. 

The first condition evidently yields a positive value ol ^ Binot it 



GCO 



APPENDIX. 



causes the first term of the preceding equation to vanish; and the second 
terra is essentially positive, a being always greater than unity. 

If, therefore, the first condition be possible, or if there be any value of 
which satisfies it, that value corresponds to a position of minimum pres 
sure. Solving, in respect to cos. 0, we obtain 



* nv+p 



2gah 



= cos. (46). 



The first condition will therefore yield a position of minimum [ires- 
sure, if 

s /(k i + l)(g + a^)(oi i -l) > — 1 8 /(ff+Z a )(y + a< o v )ia'— 1) <(a + l) 

-V 2gah~~ < + l, qfll V~ 2gah >(a— 1] 



or if 

(g + P)(y + flto'X»'--l) <(a + l) J (Z^+n(^ + «co 2 )(a--l) 

2gah >(a-l)\ orlt 2gnh{a+lf 



^ (P + Zp(y^ )(a + 1) 
2gah{a — 1)' 



(47), 



or if 



and 



A , ^ 2ga7iU+iy , 2^a + 1)' ? 



2gah(a — l)- 1 
W + Tjl^+l) 



igH*— i) 7 <7 



(tf + Z')(a+l) «' 
whence, substituting for a and reducing, we obtain finally, the conditions 

(g\ {*+(•+«•}•;_ (V)and M '>W 4^+C^ — *)']' ff\,J 

w < W(i , +20{* , +(«~ *W W > WG&M-FMfcH (*+*)} ~~W- (4a) 

Of these inequalities the second always obtains, because 

{V + (a — h)'\* < (F + P) \k' + (a + hf\ } 

whatever be the values of k, a and h. And the first is always possible, 
since 

\V+(a+hy\ *>{V+T) \V + (a — h)\. 

If the Jirst obtain, there are two corresponding positions of CA on either 
eide of the vertical, determined by equation (46.), in which the pressure Y 
of the cylinder upon the plane is a minimum. 



ROLLING MOTION OF A CYLINDER. 667 



Substituting the other two values (* and 0) of 6 which cause -r- to 

a) 

d l Y 
vanish in ths value of -,-rwe obtain the values 

Sh_{]^+l){g-\-a^){u~V)i ch (F + Z 8 )( fl r + o«o*)(a+])/ 



and 



a 2ga\a + lf ) la 2ga'(a—iy $' 



la 



or 



~al 2gha(*+\y \ al 2^a(a— l) a $'• — l 4 ** 

which expressions are both negative if the inequalities (47.) obtain. The 
same conditions which yield minimum values of Y in two corresponding 
oblique positions of CA, yield, therefore, maximum values in the two ver- 
tical positions ; so that if the inequalities (48.) obtain, there are two posi- 
tions of maximum and two of minimum pressure. 

Substituting the values of cos. (equation 46.) in equation (44.), and 
reducing, we obtain for the minimum value of Y in the case in which the 
inequalities (48.) obtain, 

Y=— $2 (a*-tf—h*)^¥+l t )(l + —\ 
4a { { \ g J 



+ 3 y(V + l){V + (a + Ji) \ \tf+(ar-hy\ (l + — \ I. 

If this expression be negative the cylinder will jump. 
In the case in which w = 0, which is that of a pendulum having a cylin- 
drical axis of finite diameter, it becomes 

Y=^-A 2a 2 — 2A 2 — 3F — Z 2 + 3 V(l<? + l*){F + (a + hy\{Jc' + (a-hy\l*....(on). 

If the first of the inequalities (48.) do not obtain, no position of mini- 
mum pressure corresponds to equation (46.) ; and the inequalities (47.) do 

d'Y 
not obtain, so that the values (49.) of -j-~, given respectively by the sub- 
stitution of 7t and for 0, are no longer both negative, but the second only. 
In this case the value n of is that, therefore, which corresponds to a posi- 
tion of minimum pressure, which minimum pressure is determined by 
substituting rt for in equation (35.), and is represented by 

1-> W 4^~ r (-TIT'S 

_W( {V +V){g+a»*) \_JN\ h(W + l')(g + aS) 



* When the pendulum oscillates on knife-edges a = 0, and this expression 
assumes the form of a vanishing fraction, whoso value may be determined by 

the known rules. See the next articla 



GQS APPENDIX. 



_AV { h {#+ (a + hy — 4aA C os.*V I (g+ a^ 1 ) { 

' ' £' + (« + A)* ) ) 

,.y=av)i-^ + K » 9 \. {....con 

The cylinder will jump if this expression be negative, that is, if 

\ >> 1 

>^' ) 4aAcos 2 9 9, 

Xf\ >1+ e-Ha+ky 



v y y , or if — < 

^ " k' + {a + hy g 



*"V i _l ^ y ^ , or if - 

~ >1+ *' + (« + A)' g l l + [a 

or, substituting and reducing, if 

g ) 4A (a + A) cos. 2 -0, / 
" > V ) 1 j _ I * 

If the angular velocity w be assumed to be that acquired in the highest 
position of the centre of gravity, 0i=rt, and cos. -■ 0, = 0. In this case, 
therefore (equation 51.) 

Y=w( 1 -^);...( 52) . 
and there will be a jump if o ! > v . . . (53). 

Tht Pendulum oscillating on Knife-edges. 

In this case a is evanescent, and to=0. Equations (31.) and (33.) 
become, therefore, 

, r 2aA(cos. — cos. fl.) nxT ^Asin.9 
-M = »=— : — 7-; and JN — 



& + h* *»+A*' 

Substituting these values of M and N in equation (30.), 

w ** ( ) 

= k ! + A' ] — 2 ^ cos " 6 — cos " °^ sin ' d — cos * 6 sin * ° \ » 

_ ^ WA' < ) 

Y= W + ^rTfr ] 2(cos. — cos. 0,) cos. — sin. 2 tf £ ; 



WA 2 
.*. X = ^TT« ( 2 cos - °i — 3 C03 - a ) sin - • • • C 54 )- 






Y 



ROLLING MOTION OF A CYLINDER. f)G9 

IMTV 008 " 2 603.0 008.0, + -J. . . . (55), 



Y is a minimum when cos. 9 = -cos. 0„ in which case 



_ TO /P i 2 



>V /* /AG 1 „ \ 

There will therefore be a jump of the pendulum upon its bearings at 
each oscillation if the amplitude 0, of the oscillation be such, that 

-cos. 8 i >^,orcos. 2 1 >??. 

8 fa Jy 



The Jump of the falsely '-balanced Carriage-wheel. 

The theory of the falsely-balanced carriage-wheel differs from that of 
the rolling cylinder, — 1st, in that the inertia of the carnage applied at its 
axle influences the acceleration produced by the weight of the wheel, as 
its centre of gravity descends or ascends in rolling; and, 2ndly, in that 
the wheel is retained in contact with the plane by the weight of the car- 
riage. The first cause may be neglected, because the displacement of the 
centre of gravity is always in the carriage-wheel very small, and because 
the angular velocity is, compared with it, very great. 

If W, represent that portion of the weight of the carriage which must 
be overcome in order that the wheel may jump (which weight is supposed 
to be borne by the plane), and if Y, be taken to represent the pressure 
upon the plane, then (equation 52.) 

Y 1 =W 1 +Y=W I +W^1- ■).--■ (57). 

In order that there may be a jump, this expression must be negative, 
or 

i>W + W, or «P> J(l + ^') , . . . . (58). 



W^» 



or 



»5(i4')--w 



610 APPENDIX. 



77ie Driving- Wheel of a Locomotive Engine. 

The attention of engineers was some years since directed to the effects 
which might result from the false balancing of a wheel by accidents on 

railway-, which appeared to be occasioned by a tendency to jump in the 
driving-wheels of the engines. The cranked axle in all case* destroys the 
balance of the driving-wheel unless a counterpoise be applied ; at that time 
there was no counterpoise, and the axle was so cranked as to displace the 
centre of gravity more than it does now. Mr. George Heatox, of Bir- 
mingham, appears to have been principally instrumental in causing the 
danger of this false-balancing of the driving-wheels to be understood. By 
means of an ingenious apparatus*, which enabled him to roll a falsely- 
balanced wheel round the circumference of a table with any given velocity, 
and to make any required displacement of the centre of gravity, he showed 
the tendency to jump, produced even by a very small displacement, to be 
so great, as to leave no doubt on the minds of practical men as to the 
danger of such displacement in the case of locomotive engines, and a coun- 
terpoise is now, I believe, always applied. To determine what is the 
degree of accuracy required in such a counterpoise, I have calculated from 
the preceding formula} that displacement of the centre of gravity of a 
driving-wheel of a locomotive-engine, which is necessary to cause it to 
jump at the high velocities not unfrequently attained at some parts of the 
journey of an express train; from such information as I have been able to 
obtain as to the dimensions of such Avheels, and their weights, and those 
of the engines f. The weight of a pair of driving-wheels, six feet in 
diameter, with a cranked axle, varies, I am told, from 2£ to 3 tons ; and 
that of an engine on the London and Birmingham Railway, when filled with 
water, from 20 to 25 tons. If n represent the number of miles per hour 
at which the engine is travelling, it may be shown by a simple calculation, 

22 n 
that the angular velocity, in feet, of a six-feet wheel is represented by - - ' 

4-t) 

or by -n very nearly. In this case we have, therefore, — since "W represents 

a 

the weight of a single wheel and its portion of the axle, and AY, represents 
the weight, exclusive of the driving-wheels, which must be raised that 



* This apparatus was exhibited by the late Professor Cowper to illustrate his 
Lectures on Machinery at King's College. It has also been placed by General 
Moiun among the apparatus of the Conservatoire des Arts et Metiers at Paris. 

\ I have not included in this calculation the inertia of the crank rods, of the 
slide gearing, or of the piston and piston rods. The effect of these is to increase 
the tendency to jump produced by the displacement of the centre of gravity 
of the wheel ; and the like effect is due to the thrust upon the piston rod 
The discussion of these subjects does not belong to my present paper. 



ROLLING MOTION OF A CYLINDER. 



671 



either side of the engine may jump* that is. half the weight of the engine 
exclusive of the driving-wheels,— W= 1± to H tons, W, = 8f to 11* tons, 

" =-«i £ = 32-19084 whence I have made the following calculations from 
formula (59.). 



1 Weight of 
itbe engine in 
tons, includ- 
i Ing the driv- 
i i;)g wheels. 

1 


Weight of a 
pair of wheels 
with cran-ked 

axle, in tons. 


Formula (59.) 
reduced. 

m «*(i + £) 


Displacement of the centre of gravity 

of a six-feet driving-wheel which 

will cause a jump of the wheel on 

the rail. 


Eate of travelling in miles per hour. 


ni 


50. 63. 


TO. 


20 


2o 
3 


1030-08 
n- 

858-4 
n- 


4128 
•3434 


•2Su7 
•2384 


•2106 
•1751 


25 


25 
3 


1287.6 

n- 

1073 
n- 


•5150 
•4292 


•3576 
•2908 


•2628 
•2189 



It appears, "by formula (59.), that the displacement of the centre of 
gravity necessary to produce a jump at any given speed, is not dependent 
on the actual weight of the engine or the wheels, hut on the ratio of their 
weights ; and, from the above table, that when the weight of the engine 
and wheels is 6§- times that of the driving-wheels, a displacement of 2f- 
inches in the centre of gravity is enough to create a jump when the train 
is travelling at sixty miles an hour, or of two inches when it is travelling 
at seventy miles; this displacement varying inversely as the square of the 
velocity is less, other things being the same, as the square of the diameter 
of the wheel is less; for the radius of the wheel being represented by a, 

the angular velocity is represented by to = "^-, and substituting this value, 

X'iQ/ 

formula (59.) becomes 



■»© 



ga- 



( 1 + w) 



* It trill be observed, that the cranks beiug placed on the axle al righl 
to one another, when the centre of gravity on the one lide is in a favourable no 



APPENDIX 



If the weight W of the wheel be supposed to vary as the square of ia 
diameter and be represented by pa% this formula will beoome 



-(«•*?) 



still showing the displacement of the centre of gravity necessary to pro- 
duce a jump to diminish with the diameter of the wheel. The>e conclu- 
sions .are opposed to the use of light engines and small driving- wheels; 
and they show the necessity of a careful attention to the true balancing of 
the wheels of the carriages as well as the driving-wheels of the engine. 
It does not follow that every jump of the wheel would be high enough to 
lift the edge of the flange off the rail ; the determination of the height of 
the jump involves an independent investigation. Every jump nevertheless 
creates an oscillation of the springs, which oscillation will not of necessity 
be completed when the jump returns ; but as the jumps are made alter- 
nately on opposite sides of the engine, it is probable that they may, and 
that after a time they will, so synchronise with the times of the oscillations, 
as that the amplitude of each oscillation shall be increased by every jump, 
and a rocking motion be communicated to the engine attended with 
danger. 

"Whilst every jump does not necessarily cause the wheel to run off the 
rail, it nevertheless causes it to slip upon it, for before the wheel jumps 
it is clear that it must have ceased to have any hold upon the rail or any 
friction. 

The Slip of the Wheel 

If f be taken to represent the coefficient of friction between the surface 
of the wheel and that of the rail, the actual friction in any position of the 
wheel will be represented by Y,/. But the friction which it is neee— ary 
the rail should supply, in order that the rolling of the wheel maybe main- 
tained, is X. It is a condition therefore necessary to the wheel not slip- 
ping that 

Y,/>X, or/ >*.... (GO). 

X 

It therefore, taking the maximum value of* in any revolution, we find 

1 1 

that/exceeds it, it is certain that the wheel cannot have slipped in that 
revolution; whilst if, on the other hand. /falls short of it. it must have 



6ition for jumping, it is in an unfavourable position on the other Bide, so that 
it can only jump on one side at once, and the efforts on the two sides aiternar* 



ROLLING MOTION OF A CYLINDER. 673 

slipped* The positions between which the slipping will take place con- 
tinually, are determined by solving, in respect to cos. 0„ the equation 

/= T ....(61). 

The application of these principles to the slip of the carriage-wheel is 
rendered less difficult by the fact, that the value of h is always in that case 

so small, as compared with the values of Tc and a, that -may be neglected 

a 
in formulas (34.) and (35.), as compared with unity. Those equations 
then become 



X 

and 



_TOsin I _0j V(g±*J)\ 

Y=5?i <*-cos. 9 + <* + aaW )"*' e l=w J 1 i ^ cos - g I 

b we obtain 

T 1= W, + w|l + ^|^f....(63). 



and 
X a] Vf + rf)f' Hn - _ ^V W+Z)\ ... (64). 

Assume 
„ /, WA g , sin.fl 

. ^__ l+)3cos. #%__ { — )3 (|3 + cos. 6) + 2 (1 + 18 cos. 0)} sin, fl 
" de (j3 + cos.tf) 2 d& (j3 + cos.0) 3 

Now if j3> 1, there will be some value of for which -= + cos. 6 = 0, and 

therefore 1-H3 cos. = 0: and since for this value of 0, — = 0, and — - 

do do* 



* Of course, the slipping, in the case of the driving-wheels of a locomotive, 
is diminished by the fact, that whilst one wheel is not biting upon the rail 
the other ia. 

43 



074: AI'I'KNDIX. 

0' 



= — 3i ,x 3 it follows that it corresponds to a maximum value of w, and 

therefore of ±. 
Y, 

But if |3 < 1, then there is some value of cos. $ for which 3 + cos. = 0, 
and therefore for which u = infinity, which value corresponds therefore in 

this case to the maximum of ±_ . 
Y, 

Thus then it appears that according as 

X 1 

the maximum value of — is attained when cos. 8 = — j3or = — ^ ; that 

is, when 

cos.0=— JHl + Yl)or = — __^!_. . . . (66). 



#-) 



X 

In the one case the maximum value of y will be infinity, .... (67). 

and in the other case it will be represented by the formula 

In the first case, i. e. when j8 < 1, the wheel will slip every time that it 
revolves, whatever may be the value of f. In the second case, or when 
3 > 1, it will slip if f do not exceed the number represented by formula 
(68.). The conditions (65.) are obviously the same with those (59.) which 
determine whether there be a jump or not, which agrees with an obser- 
vation in the preceding article, to the effect, that as the wheel must cease 
to bite upon the rail before it can jump, it must always slip before it 
can jump. "When the conditions of slipping obtain, one of the wheels 
always biting when the other is slipping, and the slips of the two wheels 
alternating, it is evident that the engine will be impelled forwards, at 
certain periods of each revolution, by one wheel only, and at others, by 
the other wheel only ; and that this is true irrespective of the action of 
the two pistons on the crank, and would be true if the steam were thrown 
off. Such alternate propulsions on the two sides of the train cannot but 



DESCENT UPON INCLINED PLANE. 675 

communicate alternate oscillations to the buffer-springs, the intervals 
between which will not be the same as those between the propulsions ; 
but they may so synchronise with a series of propulsions as that the 
amplitude of each oscillation may be increased by them until the train 
attains that fish-tail motion with which railway travellers are familiar. 
It is obvious that the results shown here to follow from a displacement of 
the centres of gravity of the driving-wheels, cannot fail also to be pro- 
duced by the alternate action of the connecting rods at the most favorable 
driving points of the crank and at the dead points,* and that the operation 
of these two causes may tend to neutralize or may exaggerate one another. 
It is not the object of this paper to discuss the question under this point 
of view. 



NOTE F. 

On the Descent upon an Inclined Plane of a Body subject to Varia- 
tions of Tempeeatuee, and on the Motion of Glaolers. 

If we conceive two bodies of the same form and dimensions (cubes, for 
instance), and of the same material, to be placed upon a uniform horizon- 
tal plane and connected by a substance which alternately extends and 
contracts itself, as does a metallic rod when subjected to variations of 
temperature, it is evident that by the extension of the intervening rod 
each will be made to recede from the other by the same distance, and, 
by its contraction, to approach it by the same distance. But if they be 
placed on an inclined plane (one being lower than the other) then when 
by the increased temperature of the rod its tendency to extend becomes 
sufficient to push the lower of the two bodies downwards, it will not have 
become sufficient to push the higher upwards. The effect of its exten- 
sion will therefore be to cause the lower of the two bodies to descend 
whilst the higher remains at rest. The converse of this will result from 
contraction; for when the contractile force becomes sufficient to pull the 
upper body down the plane it will not have become sufficient to pnll 
the lower up it. Thus, in the contraction of the Bubstanoe whirl, inter- 
venes between the two bodies, the lower will remain at rest whilst the 
upper descends. As often, then, as the expansion and oontraotion is 
repeated the two bodies will descend the plane until, step by step, they 
reach the bottom. 



* A slip of the wheel may thus be, and probably is, produced el eaeb rare 

lution. 




G*u APPENDIX. 

Suppose the uniform bar AB placed on an inclined plane, and subjecl 
to extension from increase of temperature, a por- 
^t^y^ tion XB will descend, and the rest XA will ascend; 
the point X where they separate being determined 
by the condition that the force requisite to push 
XA up the plane is equal to that required to push 
XB down it. 

Let AX = #, AB = L, weight of each linear unit = jt, c, = inclination 
of plane, $ = limiting angle of resistance. 

.'./xx = weight of AX. ' 

^(L— <c) = BX. 

Now, the force acting parallel to an inclined plane which, is necessary 

to push a weight W up it, is represented by W — — - — ; and that ne- 



cessary to push it down the plane by W — '■ — - — . (Art. 241. 



cos. ^ 



sin.(0 + O_ T ^ sin. ( — Q 
^ cos. <p cos. <p 

.'.x {sin. (<p + + sin. (<p — 0} = L sm - (? — 

.*.2a; sin. <p cos. t = L sin. (<p — *) 

. * it 8ip -fo— 

. • . X = * JL - ? 

sin. $ cos. i 



.-.aj = iL< 1 — \ 

( tan. $ ) 




•x 



When contraction takes place, the converse of 
the above will be true. The separating point X 
will be such, that the force requisite to pull XB up 
the plane is equal to that required to pull AX 
down it. BX is obviously in this case equal to AX 
in the other. 
Let % be the elongation per linear unit under any variation of tempera- 
ture; then the distance which the point B (fig. l.)will be made to descend 
by this elongation = \ . BX 

=X(L— a;) 



ixL 



( tan. i \ 
tan.</>y 



DESCENT UPON INCLINED PLANE. 677 

If we conceive the bar now to return to its former temperature, con- 
tracting by the same amount (x) per linear unit; then the point B 
(fig. 2.) will by this contraction be made to ascend through the space 
BX . %—xk 



1T f . tarnt ) 




Total descent I of B by elongation and contraction is therefore determined 
by the equation 

tan.* ' ' 

v^^^ To determine the pressure upon a nail driven 
through the rod at any point P fastening it to the 
plane. 

It is evident, that in the act of extension the part BP of the rod will 
descend the plane and the part AP ascend; and conversely in the act of 
contraction ; and that in the former case the nail B will sustain a pressure 
upwards equal to that necessary to cause BP to descend, and a ppessure 
downwards equal to that necessary to cause PA to ascend ; so that, as- 
suming the pressure to be downwards, and adopting the same notation as 
before, except that AP is represented by p, AB by a, and the pressure 
upon the nail (assumed to be downwards) by P, we have in the case of 
extension 

_ sin. (0 + , .sin.fo— t) 
P = W -^ ti*-?)-^-, 

and in the case of contraction, 

sin. (0 + si n. (<p— Q 

F = fli *-P ) -^1> ^ cos.* ' 

Reducing, these formulae become respectively, 

P= < 2p sin. f cos. i — a sin. {<p — \ ' ' ' ' ( 3 )- 

cos. <p { r ) 

< a sin. (0 + 1) — 2p sin. <p cos. t > * * * • (4). 



P = 



COS. (j) 



Example of tite Descent of the Lead on the Roof of Bbbmk 

CATnEDKAL. 

My attention was first drawn to the influence of variation* In temper*- 
ture to cause the descent of a lamina of metal resting on an Inclined plane 



078 APPENDIX. 

by observing, in the autumn of 1853, that a portion of the lead which 

the south side of the choir of Bristol Cathedral, which had been 

renewed in the year 1851, hut had not been properly fastened to the ridge 

beam, had descended bodily eighteen indies into the gutter; so that if 

plates of load had not been inserted at the top, a strip of the roof of that 
length would have been left exposed to the weather. The sheet of lead 
which had so descended measured, from the ridge to the gutter, 19ft. 4in., 
and along the ridge 00ft. The descent had been continually going On 
from the time the lead had been laid down. An attempt made to stop it 
by driving nails through it into the rafters had failed. The force by 
which the *lead had been made to descend, whatever it was, had been 
found sufficient to draw the nails* As the pitch of the roof was only 
10|° it was sufficiently evident that the weight of the lead alone could not 
have caused it to descend. Sheet lead, whose surface is in the state of 
that used in roofing, will stand firmly upon a surface of planed deal when 
mclined at an angle of 30 "f, if no other force than its weight tends to 
cause it to descend. The considerations which I have stated in the pre- 
ceding articles, led me to the conclusion that the daily variations in the 
temperature of the lead, exposed as it was to the action of the sun by its 
southern aspect, could not but cause it to descend considerably, and the 
only question which remained on my mind was, whether this descent 
could be so great as was observed. To determine this I took the follow- 
ing data : — 
Mean daily variation of temperature at Bristol in the 

month of August ; assumed to be the same as at Leith 

(Kcemtz Meteorology, by Walker, p. 18.) - - - 8° 21' Cent. 
Linear expansion of lead through 100° Cent. - - - -0028436. 
Length of sheets of lead forming the roof from the ridge 

to the gutter 232 inches. 

Inclination of roof 16° 32'. 

Limiting angle of resistance between sheet lead and deal - 30° 

Whence the mean daily descent of the lead, in inches, in the month ol 
August, is determined by equation (2.) to be 



* The evil was remedied by placing a beam across the rafters, near the ridge, 
and doubling the sheets round it, and fixing their ends with spike-nails. 

f This may easily be verified. I give it as the result of a rough experiment 
of my own. I am not acquainted with any experiments on the friction of lead 
male with sufficient care to be received as authority in this matter. The 
friction of copper on oak has, however, been determined by General Morin 
(see a table in the preceding part of this work N i to be 0*6'2, and its limiting angle 
of resistance 31° 48' ; so that if the roof of Bristol Cathedral had been inclined 
at 31° instead of 16°, and had been covered with sheets of copper resting on 
oak boards, instead of sheets of lead resting on deal, the sheeting would not 
have slipped by its weiyht only. 



DESCENT UPON INCLINED PLANE. 679 



, nnn 8-21 tan. 16° 32' 

Z=232 x — — x -0028436 X 



100 tan. 30 

1= -027848 inches. 

This average daily descent gives for the whole month of August a descent 
of -863288. If the average daily variation of temperature of the month 
of August had continued throughout the year, the lead would have 
descended 10-19148 inches every year. And in the two years from 
1851 to 1853 it would have descended 20*38296 inches. But the daily 
variations of atmospheric temperature are less in the other months of the 
year than in the month of August. For this reason, therefore, the cal- 
culation is in excess. For the following reasons it is in defect : — 1st., 
The daily variations in the temperature of the lead cannot but have been 
greater than those of the surrounding atmosphere. It must have been 
heated above the surrounding atmosphere by radiation from the sun in 
the day-time, or cooled below it by radiation into space at night. 2ndly., 
One variation of temperature only has been assumed to take place every 
twenty-four hours, viz. that from the extreme heat of the day to the 
extreme cold of the night ; whereas such variations are notoriously of 
constant occurrence during the twenty-four hours. Each cannot but have 
caused a corresponding descent of the lead, and their aggregate result 
cannot but have been greater than though the temperature had passed 
uniformly (without oscillations backwards and forwards) from one extreme 
to the other. 

These considerations show, I think, that the causes I have assigned are 
sufficient to account for the fact observed. They suggest, moreover, the 
possibility that results of importance in meteorology may be obtained 
from observing with accuracy the descent of a metallic rod thus placed 
upon an inclined plane. That descent would be a measure of the aggre- 
gate of the changes of temperature to which the metal was subjected 
during the time of observation. As every such change of temperature is 
associated with a corresponding development of mechanical action under 
the form of work,* it would be a measure of the aggregate of Buoh changes 
and of the work so developed during that period. And relations might be 
found between measurements so taken in different equal periods of time 
■ — successive years for instance — tending to the development of new 
meteorological laws. 



* Mr. Joule has shown (Phil. Trans , 1850, Part I.) that the quantity of beat 
capable of raising a pound of water by 1° Pah, requirea for its evolution 11% 
units of work. 



6S0 APPENDIX 



The Descent of Glaciers. 

The following are the results of recent experiments * on the expansion 
of ice : — 

Linear Expansion of Ice for an Interval of 100° of the Centigrade 

Thermometer. 

0-00524 Schumacher. 

000513 Pohrt. 

0-00518 Moritz. 
Ice, therefore, has nearly twice the expansibility of lead; so that a 
sheet of ice would, under similar circumstances, have descended a plane 
similarly inclined, twice the distance that the sheet of lead referred to in 
the preceding article descended. Glaciers are, on an increased scale, 
sheets of ice placed upon the slopes of mountains, and subjected to 
atmospheric variations of temperature throughout their masses by varia- 
tions in the quantity and the temperature of the water, which, flowing 
from the surface, everywhere percolates them. That they must from this 
cause descend into the valleys, is therefore certain. That portion of the 
Mer de Glace of Chamouni which extends from Montanvert to very near 
the origin of the Glacier de Lechaud has been accurately observed by 
Professor James Forbes.t Its length is 22,600 feet, and its inclination 
varies from 4° 19' 22" to 5° 5' 53". The Glacier du Geant, from the 
Tacul to the Col du Geant, Professor Forbes estimates (but not from his 
own observations, or with the same certainty) to be 24,700 feet in length, 
and to have a mean inclination of 8° 46' 40". 

According to the observations of De Saussure, the mean daily range 
of Reaumur's thermometer in the month of July, at the Col du Geant, is 
4 0, 257}, and at Chamouni 10 o, 092. The resistance opposed by the 
rugged channel of a glacier to its descent cannot but be different at dif- 
ferent points, and in respect to different glaciers. The following passage 
from Professor Forbes's work contains the most authentic information I 
am able to find on this subject. Speaking of the Glacier of la Brenva 
he says : — " The ice removed, a layer of fine mud covered the rock, not 
composed, however, alone of the clayey limestone mud, but of sharp sand 
derived from the granitic moraines of the glacier, and brought down with 
it from the opposite side of the valley. Upon examining the face of the 
ice removed from contact with the rock, we found it set all over with 
sharp angular fragments, from the size of grains of sand to that of a 
cherry, or larger, of the same species of rock, and which were so firmly 



* Vide Arehir. i. Wissenschaftl. Kunde v. Russland, Bd. vii s. 888. 
f Travels through the Alps of Savoy. Edinburgh, 1S53. 
% Quoted by Professor Forbes, p. 231. 



DESCENT OF GLACIERS. 681 

fixed in the ice as to demonstrate the impossibility of such a surface being 
forcibly urged forwards without sawing any comparatively soft body 
which might be below it. Accordingly, it was not difficult to discover in 
the limestone the very grooves and scratches which were in the act of 
being made at the time by the pressure of the ice and its contained frag- 
ments of stone." (Alps of the Savoy, pp. 203 — 4.) It is not difficult 
from this description to account for the fact that small glaciers are some- 
times seen to he on a slope of 30° (p. 35.). The most probable supposition 
would indeed fix the limiting angle of resistance between the rock and 
the under surface of the ice set all over, as it is described to be, with 
particles of sand and small fragments of stone, at about 30°; that being 
nearly the slope at which smooth surfaces of calcareous stone will rest on 
one another. If we take then 30° to be the limiting angle of resistance 
between the under surface of the Her de Glace and the rock on which it 
rests, and if we assume the same mean daily variation of temperature 
(4*257 Reaumur, or 5*321 Centigrade) to obtain throughout the length 
of the Glacier du Geant, which De Saussure observed in July, at the 
Col du Geant; if, further, we take the linear expansion of ice at 100° 
Centigrade to be that (-00524) which was determined by the experiments 
of Schumacher, and, lastly, if we assume the Glacier de Geant to descend 
as it would if its descent were unopposed by its confluence with the 
Glacier de Lechaut; we shall obtain, by substitution in equation (2.) 
for the mean daily descent of the Glacier du Geant at the Tacul, the 

formula 

•00524 tan- 8° 46' 
Z= 24700 x 5-321 X-—X ^^-3^- 

1= 1-8395 feet 

The actual descent of the glacier in the centre was 1*6 feet. If the 
Glacier de Lechaut descended, at a mean slope of 5°, singly in a sheet of 
uniform breadth to Montanvert without receiving the tributary glacier of 
the Talefre, or uniting with the Glacier du Geant, its diurnal descent would 
be given by the same formula, and Avould be found to be -954S7 feet 
Reasoning similarly with reference to the Glacier du Geant; supposing it 
to have continued its course singly from the Col du Geant to Montanvert 
without confluence with the Glacier de Lechaut, its length being 40,480 
feet, and its mean inclination 6° 53', its mean diurnal motion I at Montan- 
vert would, by formula (2.) have been 2-3564* feet. The actual mean 
daily motion of the united glaciers, between the 1st and the 28th July, * BB 
at Montanvert,t 



* On the 1st of July the centre of the actual motion of the Mor d« QlMf at 
Montanvert was 2*25 feet. 

f Forbes' " Alps of Savoy," p. 140. 



6S2 APPENDIX. 

Near the aide of the glacier - - 1*441 feet 
Between the side and the centre - 1*750 " 
Near the centre 2-141 M 

The motion of the Glacier <le Lechant was therefore accelerated by their 
contluence, and that of the Glacier du Geant retarded. The former is 
__< «1 down by the latter. 

I have had the less hesitation in offering this solution of the mechanical 
problem of the motion of glaciers, as those hitherto proposed ar 

38 y imperfect. That of De Saussure. which attributes the descent of 
the glacier simply to its weight, is contradicted by the fact that isolated 
fragments of the glacier stand firmly on the slope on which the whole 
neverti * scends. It being obvious that if the parts would remain at 
arately on the bed of the glacier, they would also remain at rest 
when united. 

That of Professor J. Forbes, which supposes a viscous or semi-fluid 
structure of the glacier, is not consistent with the fact that no viscosity is 
to be traced in its parts when separated. They appear as solid fragments, 
and they cannot acquire in their union properties in this respect which 
individually they have not. 

Lastly, the theory of Charpentier, which attributes the descent of the 
glacier to the daily congelation of the water which percolates it. and the 
expansion of its mass consequent thereon, whilst it assigns a cause which, 
so far as it operates, cannot, as I have shown, but cause the glacier to 
descend, appears to assign one inadequate to the result : for the congelation 
of the water which percolates the glacier does not, according to the obser- 
vations of Professor Forbes.* take place at all in summer more than a few 
inches from the surface. Nevertheless, it is in the summer that the daily 
motion of the glacier is the greatest. 

The following remarkable experiment of Mr. Hopldns of Cambt 
which is con-idered by him to be confirmatory of the sliding theory of 
De Saussure as opposed to De Charpentier's dilatation theory, rf 
a ready explanation on the principles which I have laid down in this 
note. It is indeed a necessary result of them. Mr. Hopkins placed a 
as of rough ice, confined by a square frame or bottomless box, upon 
a roughly chiselled flag-stone, which he then inclined at a small angle; 
and found that a slow but uniform motion was produced, when even it 
was placed at an inconsiderable slope. This motion, which Mr. Hopkins 
attributed to the dissolution of the ice in contact with the stone, would, 
I apprehend, have taken place if the mass had been of lead instead of ice ; 



* "Travels in the Alps," p. 413. 

f I have quoted the following account of it from Professor Forbes's book, 
p. 419. 






DIMENSIONS OF A BUTTRESS. 083 

and it would have been but about half as fast, because the linear expan- 
sion of lead is only about half that of ice. 



NOTE G. 
The best Dimensions or a Buttress. 

If m, (Art. 299.) represent the modulus of stability of the portion AG cf 
the wall, it may be shown, as before, that 

P{(A t — ^ 2 )sin. a — (I — a 2 — w^cos. a} = (|a, — m l ){h l — AJa,^; 

. * . F{(h x — h 2 )sm. a — (I — a 2 )cos. a} 

= K^i — h 2 )a x V — m i{? cos. a + (K — h^aifi} 

If m l =w, the stability of the portion AG of the structure is the same 
with • that of the whole AC ; an arrangement by which the greatest 
strength is obtained with a given quantity of material (see Art. 388.). 
This supposition being made, and m eliminated between the above equa- 
tion and equation (388.), that relation between the dimensions of the 
buttress and those of the wall which is consistent with the greatest 
economy of the material used will be determined. The following is that 
relation : — 

^piafhi + 2tf !&A + -a 2 h. 2 ) — P (hi sin. a — I cos. a) 

7h 



P cos. a + /*(a A + ~a 2 h 2 ) 

_ \^Jh — h 2 )a? — P{(^,— A 2 ) sin.a — (I— Qcos.a} 
P cos. a + pa>i (K — h 2 ) 

It is necessary to the greatest economy of the material of the Gothic 
buttress (Art. 301.) that the stability of the portions Qa and Q&, upon 
their respective bases ac and 5<s, should be same with that of the whole 
buttress on its base EC. If, in the preceding equation, h x — //, be 
substituted for A„ and h 2 — h 3 for A„ the resulting equation, together 
with that deduced as explained in the conclusion of Art. 801., will deter- 
mine this condition, and will establish those relatione between the dimen- 
sions of the several portions of the buttress which are consistent with the 
greatest economy of the material, or which yield the greatest Btrength to 
the structure from the use of a given quantity of material. 



6%± 



APPENDIX. 



NOTE H. 

Dimensions of tiie Teeth of Wheels. 

The following rules are extracted from the work of M. Morin, entitled 
Aide Mernoire de Jlecanique Pratique : — If we represent by a the width 
in parts of a foot of the tooth measured parallel to the axis of the wheel, 
and by 5 its breadth or thickness measured parallel to the plane oi 
rotation upon the pitch circle; then, the teeth being constantly greased, 
the relation of a and 5 should be expressed, when the velocity of the pitch 
circle does not exceed 5 feet per second, by a = 45 ; when it exceeds 
5 feet per second, by a = 55 : if the wheels are constantly exposed to wet, 
by a = 65. 

Tfrese relations being established, the width or thickness of the tooth 
will be determined by the formulas contained in the columns of the follow- 
ing table: — 



Material 


French measures, cents 
and kils. 


English measures, feet 
and pounds. 


Cast iron ... 
Brass ... 
Hard wood 


6=-105 4 / P 

6=131 fP 

5= 145 v'P 


6 = 002319 tT 
b =-002894 i P 
&=-003203 S Y 

i 



Assuming that when the teeth are carefully executed the space between 
the teeth should be y 3 th greater than their thickness, and -^th greater 
when the least labor is bestowed on them, the values of the pitch T will 
in these two cases be represented by 5(2 + T i) an( l ^(2 + yV), or by 20t'»75 
and 2'15. Substituting in these expressions the values of 5 given by the 
formulas of the preceding table, then determining from the resulting 
values of c (see equation 233.) the corresponding values of the coefficient 
C (see equation 234.), the following table is obtained : — 



Material. 


Yalue of c (equation 288. . 


Value of C (equation 234.). 


For teeth of For teeth of 
the best work- inferior work- 
manship, manship. 


For teeth of 
the best work- 
manship. 


For teeth of 
inferior work- 
manship. 


Cast iron - 
Brass 
Hard wood 


1 E - 
■006621 


•004 i 

•(•06077 

-006726 


0912 

1-087 

1-131 


0-922 
1-068 
1143 



TRACTION OF CARRIAGES. 6 $5 

The following are the pitches commonly in use among mechanics : — 

in. in. in. in. in. in. in. 

1, H, li, H, 2, 21, 3. 

Prof. Willis considers the following to be sufficient below inch pitch :— 
in. in. in. in. in. 

i, i, h f, *• 

Having, therefore, determined the proper pitch to be given to the tooth 
from formula 234., the nearest pitch is to be taken from the above series 
to that thus determined. 



NOTE I. 

Experiments of M. Morin on the Traction of Carriages. 

The following are among the general results deduced by M. Morin from 
his experiments : — 

1. The traction is directly proportional to the load, and inversely pro- 
portional to the diameter of the wheel. 

2. Upon a paved or a hard Macadamised road, the resistance is independ- 
ent of the width of the tire when it exceeds from 3 to 4 inches. 

3. At a walking pace the traction is the same, under the same circum- 
stances, for carriages with springs and without them. 

4. Upon hard Macadamised and upon paved roads the traction increases 
with the velocity ; the increments of traction being directly proportional 
to the increments of velocity above the velocity 3*28 feet per second, or 
about 2i miles per hour. The equal increment of traction thus duo to 
each equal increment of velocity is less as the road is more smooth, and 
the carriage less rigid or better hung. 

4. Upon soft roads of earth, or sand or turf, or roads fresh and thickly 
graveled, the traction is independent of the velocity. 

5. Upon a well-made and compact pavement of hewn stones the traction 
at a walking pace is not more than three-fourths of that upon the best 
Macadamised road under similar circumstances; at a trotting paoe it 
is equal to it. 

6. The destruction of the road is in all cases greater as the diametePB 
of the wheels are less, and it is greater in carriages without than with 
springs. 



APPENDIX. 

NOTE K. 

Ox the Strength of Columxs. 

Mr. Hodgktxson has obligingly communicated the following observations 
on Art. 430. :— 

1. The reader must be made to understand that the rounding of the 
ends of the pillars is to make them moveable there, as if they turned by 
means of a universal joint : and the llat-ended pillars are conceived to be 
supported in every part of the ends by means of flat surfaces, or otherwise 
rendering the ends perfectly immoveable. 

2. The coefficient (13) for hollow columns with rounded ends is deduced 
from the whole of the experiments first made, including some which were 
very defective on account of the difficulty experienced in the earlier 
attempts to cast good hollow columns so small as were wanted. The 
first castings -were made lying on their side : and this, notwithstanding 
every effort, prevented the core being in the middle : some of the columns 
were reduced, too. in thickness, half way between the middle and the 
ends, and near to the ends, and this slightly reduced the strength. These 
causes of weakness existed much more among the pillars with rounded 
ends than those with flat ones : they are alluded to in the paper (Art 
Had it not been for them, the coefficient (13) would, I conceive, have 
been equal to that for solid pillars (or 14-9). 

3. The fact of long pillars with flat ends being about three times as 
strong as those of the same dimensions with rounded ends is, I conceive. 
well made out, in cast iron, wrought iron, and timber; you have, how- 
ever, omitted it. being perhaps led to do it through the low value of the 
coefficient (13) above mentioned. 

The same may be mentioned with respect to the near approach in 
strength of long pillars with flat ends, and those of half the length with 
rounded ends. It may be said that the law of the 1*7 power of the length 
would nearly indicate the latter ; but this last, and the other powers 
and 3*55. are only approximations, and not exactly constant, though 
nearly so, and I do not know whether the other equal quantities are not, 
with some slight modifications, physical facts. 

4. The strength of pillars of similar form and of the same materials 
varies as the 1*865 power, or near as the square of their like linear 
dimensions, or as the area of their cross section. 



COMPLETE ELLIPTIC FUNCTIONS. 



687 



TABLE I. 
The Numerical Values o/ 1 complete Elliptic Functions of the first and 
second Orders for Values of the Modulus k corresponding to each Degree 
of the Angle sin.— '&. 



Sin.— i*. | 


F.. 


Ei. 


Sin.— ik. 


Fi. 


B|. 


0° 


1-57079 


1-57079 


46 


1-86914 


1 34180 


1 


1-57091 


1-57067 


47 


1-88480 


1-33286 


2 


1-57127 


1-57031 


48 


1-90108 


1-32384 


3 


1-57187 


1-56972 


49 


1-91799 


1-31472 


4 


1-57271 
1-57379 


1-56888 
1-56780 


50 


1-93558 


1-30553 
















51 
52 


1-95386 
1-97288 


1-29627 
1-28695 


6 


1-57511 


1-56649 


7 


1-57667 


1-56494 


53 


1-99266 


1-27757 


8 


1-57848 


1-56316 


54 


201326 


1-26814 


9 


1-58054 


1-56114 
1-55888 


55 


2-03471 


1-25867 


10 


1*58284 


56 

57 


2-05706 
2-08035 


1-24918 
1-23966 


11 


1-58539 


1-55639 


12 


1-58819 


1-55368 


58 


2-10465 


1-23012 


13 


1-59125 


1-55073 


59 


2-13002 


1-22058 


14 


1-59456 


1-54755 
1-54415 


60 


2-15651 


1-21105 


15 


1 -59814 














61 
62 


2-18421 
2-21319 


1-20153 
IT 9204 


16 


1-60197 


1-54052 


17 


1-60608 


1-53666 


63 


2-24354 


1-18258 


18 


1-61045 


1-53259 


64 


2-27537 


1-17317 


19 


1-61510 


1-52830 
1-52379 


65 


2-30878 


116382 


20 


1-62002 


66 
67 


2-34390 

2-38087 


1-15454 
1-14534 


21 


1-62523 


1-51907 


22 


1-63072 


1-51414 


68 


241984 


1-13624 


23 


1-63651 


1-50900 


69 


246099 


IT 27 24 


24 


1-64260 


1-50366 


70 


2-50455 


1-11837 


25 


1-64899 


1-49811 


71 

72 


2-55073 
2-59981 


1-10964 
1-10106 


26 


1-65569 


1-49236 


27 


1-66271 


1 -48642 


73 


2-65213 


1-09265 


28 


1-67005 


148029 


74 


2-70806 


1-08412 


29 
30 


1-67773 
1-68575 


1-47396 
146746 


75 


2-76806 


1-07640 


76 

77 


2-88267 

2*90256 


1*06860 
106105 


31 


1-69411 


146077 


32 


1-70283 


145390 


78 


2-97856 


1 068T7 


33 


1-71192 


1-44686 


79 


3-06172 


1*041 


34 
35 


1-72139 
1-73124 


1-43966 
1-43229 


80 


31 5338 


1*04011 


81 
82 


3-25530 
8-86986 


1*03878 
1*02784 


36 


1-74149 


142476 


37 


1-75216 


1-41707 


83 


1042 


1*02281 


38 


1-76325 


1 40923 


84 


8*65186 


1*01728 


39 
40 


1-77478 
1-78676 


1-40125 
1-39314 


85 


8*88174 


1*01266 


86 

87 


4-05275 
4*8! 


1*008 
1*00 


41 


1-79922 


1-38488 


42 


1-81215 


1-87650 




4-74271 


1*00 


43 


1-82560 


1-86799 


89 


5*48490 


I "01 


44 


1*83956 


1-35987 








45 


1-85407 


1 -3506 i 












.. 



The Tables of M. Garidel, 



TABLE LL 



Showing the Angle of Rupture ¥ of an Arch whose Loading is of ike i 

I rial with its Voussoirs, and whose Eztrados is inclined at a given 
Angle to the Horizon. - 

a = ratio of lengths of Youssoirs to radius of intrados. 

io of depth of load over crown to radius of intrados, so that 
- = 3(1 -fa). (Art 838.) 
t, = inclination of extrados to horizon. 






■ 




e=0-l 


.-=:-. 


:=:-! 


: = ■ i 


.- = : ' 


— : i 


0^05 




" - . - ' 


54v4 : 


5115 s 








010 




- 4^ 




.:■:•■:: 


54-93 






015 


64 1 


61-3 




•:v- - 


58-0 




56-21 


- 


631 


61-7 


r - 


6040 


i?-y. 


59-60 


:*-i 


■--- 




61-76 


61-44 


61-22 


61-05 


-: \<i 


6059 


o 3 :• 


61-3 


61-42 




-::-:: 


61-66 


61-67 


61-81 


0-35 


" 


■; ?: 


61-21 


61-54 


61^8 


-.: .-* 


62-56 


040 




59-8 


- - 


61-05 


1 1 -: 8 


61-67 


62-9 




n . 






6019 


-. - 


-- 




050 


5563 






:v- 


" - - 


■ U 


-_.: 



: bo' 



c 




e=0-l 


- 


e=0"S 


.= 4 


:=■■* 


c=l-0 | 




"- 




4 - 1 " 












■ ri - 


55-95 






- 


: ■■• * 




- 


' 1 


H 




"~ 


56-61 


55-66 : 


- 


-■:■-:? 


- 


r 72 


5935 


■ " 






a 


n 


" 




•:■:•-: - 


60-33 




6017 ! 


0-30 


60-09 


I 43 


n 


60-95 




-:: 11 




- 


n 


6012 




61-02 


61-33 


6159 








69-83 


6011 


' - 


-".■> 


" 




- 


- 




■ . 


r ' - 7 


60"67 


6116 




050 




57 13 




• 


59-81 


60-41 


6245 



ANGLE OF RtfPTUBE OF AX ARCH. 



GSO 



f = 15°. 



a 


c=0 


c=0-l 


c=0-2 


c=0-3 


c=0-4 


c=0-5 


( 
c— 10 


O'Oo 


64-8° 


50-5° 


46-95° 


45-69° 


45-03° 


44-67° 


43-9° 


o-io 


59-3 


55-07 


53-31 


52-47 


5199 


51-69 


50-93 


O'lo 


59-OS 


57-32 


56-65 


56-05 


55 - 75 


55*55 


55-05 


0-20 


59-06 


58-60 


58-35 


5820 


58-J0 


68-02 


57-84 


0-25 


59-05 


59-28 


59-42 


59-53 


59-60 


59-65 


59 79 


0-30 


58-90 


59-57 


59-98 


60-26 


60-48 


60-66 


61-15 


035 


58-53 


5941 


60-09 


60-57 


60 93 


61-17 


62-0 


0-40 


57-99 


59-08 


5987 


60-48 


60-95 


61-36 


62-6 


0-45 


57'2ti 


58-43 


59-34 


60-D6 


60-67 


61-15 


G2'7 


0-50 


56-38 


57-61 


58-58 


59-36 


60-06 


6064 


62-5 



. = 22° 30'. 



a 
0-05 


c=0 


c=0-l 


c=0*2 


c=0-8 


c=0-4 


c=0-5 


c=lO 


36-1° 


41-2° 


42-0° 


42-3° 


42-6° 


42-7° 


42-9° 


o-io 


50-5 


50-3 


5019 


50-17 


50-14 


50-13 


50-11 


0-15 


54*25 


54-31 


54-35 


54-35 


54-36 


54*36 


54-38 


0-20 


56-17 


56-60 


56-82 


56-95 


57-04 


57-11- 




0-25 


57-27 


57-93 


58-33 


58-61 


58*79 


58*95 


59-33 


0-30 


57 -S 5 


58*68 


59-23 


59-60 


59-93 


60-16 


60 83 


0-35 


58-07 


59-01 


59-70 


60-21 


60-61 


60-91 


61-85 


0-40 


58-02 


59-02 


59-79 


6038 


6087 


61-25 


62-2 


0-45 


57-74 


58-78 


59-60 


60-26 


60-82 


6127 


62*7 


0-50 


57-30 


58-31 


59-16 


59 88 


60-47 


61-00 


62-9 



30°. 



* 


c=0 


c=0-l 


c=0*2 


c=0-3 


c=0-4 


c=0-5 


o=l-0 


0-05 


31 3° 


36-2° 


38*4° 


39-57° 


40-28° 


40*77' 


■l 1 -9 ' 


o-io 


43-3 


46-06 


47-25 


47-90 


48 30 


48*59 




0-15 


50 07 


51-46 


52-18 


62*68 


52*94 


58*1 i 




0-20 


53-66 


54-69 


55*27 


55-67 


55*96 


56*16 




0-25 


55-80 


56-72 


57-30 


57-72 


68*01 






0-30 


57-13 


58-01 


58*62 


59*06 


59*40 






0-35 


57-93 


58-80 


59-43 




60 88 


60*68 


81*64 


0*40 


58-33 


59-20 


5989 


60*42 


60*87 


61*28 




0-45 


58 47 


59-88 


60'08 


60 61 


61*08 






0-50 


58-38 


59*22 


59-98 




61*08 







u 



600 



APPENDIX. 



37° 30'. 



a 


e=0 


c=01 


c=0-8 


c=0-3 


c=04 


c=05 


c=10 


0-05 


311° 


34-3° 


36-28° 


37-59° 




3916° 




o-io 


40-98 


43-59 




46-01 




47-14 


48 35 


0*16 


47-71 


49-40 




51-12 


51-61 


5196 


52-93 




5201 


52-23' 


64-01 


54-54 


54-94 




56-10 






55 SO 


66-45 


56 94 




57-59 


5841 


030 




W1W 


68*16 


5S-62 


58-98 


5926 


60-16 


035 


ra i 




59-34 


5981 


60-17 




61-45 


0-40 




59*58 


60-13 


60-60 


60-97 


61-30 


63-2 


45 


5938 


6006 


60-62 


61-07 


61-47 


61-83 


63-0 


0-50 


5669 


60-29 


60-84 


61-26 


61-72 


62-07 


633 



t = 45\ 



a 


c=0 


c=0-l 


c=0-8 


e=0« 


c=04 


c=06 


c=l-0 


0-05 


31-3° 


33-68° 


35-46° 


36-36° 


37-22° 


38-0° 


39-9° 


010 


40-6 


42 4 


43-7 


44-64 


45-35 


45-92 


47-45 


0-15 


46-77 


48-20 


49-18 


4993 


5047 


50-92 


52-15 


0-20 


5123 


52 27 


53-05 


53-64 


54-U7 


54 42 


55-47 


0-25 


5442 


55-22 


55-84 


56-31 


56-70 


57-01 


57-97 


0-30 


56-72 


57-38 


57-90 


58-30 


5S-65 


58-94 


59-S5 


0-35 


58-35 


58-94 


59-40 


59-79 


6011 


60-38 


61-30 


0-40 


5956 


6009 


60 52 


60-S9 


61-19 


61-46 


62-4 


0-45 


60-40 


60-89 


61-29 


61-67 


61-97 


62-24 


63-2 


050 


60-99 


6143 


61-8 


62 2 


62-5 


62-8 


63-8 



HORIZONTAL THRUST OF AN ARCH. 



691 



The Tables of M. Garidel. 



TABLE III. 



Showing the Horizontal Thrust of an Arch, the Radius of whose Intrudos 
is Unity, and the weight of each Cubic Foot of its Material and that of 
its Loading, Unity. (See Art. 344.) 

KB. To find the horizontal thrust of any other arch, multiply that given 
in the table by the square of the radius of the intrados and by the weight 
of a cubic foot of the material. 



= 0. 





c=0 


c=01 


c=0-2 


c=0-3 


c=0-4 


c=0-5 


e=l«0 


a 


P 


P 


P 


P 


P 


P 


P 




r i 


r 2 


r e 


7-2 


^T 


r2 


r- 


0-05 


0-08174 


0-14797 


0-21762 


0-28877 


0-36060 


0-43277 


0-79541 


o-io 


0-10279 


0-16370 


0-22588 


0-28862 


0-35164 


0-41481 


073161 


0-15 


0-11894 


0-17480 


0-23111 


0-28764 


0-34429 


0-40100 


0-68504 


0-20 


013073 


0-18191 


023322 


0-28460 


0-33603 


0-38747 


0*64488 


0-25 


0-13871 


0-18553 


0-23237 


0-27922 


0-32607 


037293 


0-60727 


0*30 


0-14333 


0-18604 


0-22874 


0-27145 


0-31416 


0-35687 


0-57041 


0-35 


0-14504 


0*18379 


0-22258 


0-26140 


0-30023 


0-33907 


0-5:!385 


0*40 


0-14422 


0*17913 


0-21415 


0-24924 


0-28437 


0-31953 


0-49560 


0-45 


0-14124 


0-17240 


0-20374 


0-23520 


0-26674 


0-29835 


0*45698 


0-50 


0-13649 


016396 


0-L9168 


0-21957 


0-24760 


0-27573 


417 28 



7° 30'. 





c=0 


c=0-l 


c=0-2 


c=0-8 


c=0-4 


c=0-5 


o 10 


« 


P 


P 


P 


P 


P 


P 


P 




r 2 


r« 


r' 


r2 


r 2 


r 2 


r i 


0-05 


0-06180 


0-12867 


0-19937 


0-27125 


(r:; !:;.-,(', 


0-41606 


0-7 7 '.'4 4 


o-io 


0-08514 


0-14666 


0-20930 


0-27237 


0-88561 


0-89895 


0-71618 


0-15 


0-10380 


0-16001 


0-21657 


0-27326 


0-88008 




0-67110 


0-20 


0-11813 


0-16948 


0*22089 


0*27281 


0*82884 






0-25 


0-12870 


0-17557 


0-22244 


Q-26982 


0-81619 


0*86808 




30 


0-13598 


0-17866 


0-22134 


0-26408 


0-80678 


0-84948 




0-35 


0-14040 


017909 


021783 


0-25661 


0-29642 


0-88424 




0-40 


0-14234 


0-17718 


0-21215 


0-24720 




o-:;i7i i 


0*49844 


0-45 


0-14211 


0-17323 


20454 


0-28598 


0'26751 


29910 




0-60 


0-14003 


0-16753 


019528 


0-22819 


0-25124 


0*27988 


0*42096 

j 



692 



APPENDIX. 

* = 15°. 



1 


c=0 


c=0-l 


c=0-2 


c=0-3 


c=0-4 


e=0-5 


c=l-0 


a 


P 


P 


P 


P 


P 


P 


P 




r 2 


r-2 


r -i 


r -2 


r2 


r 2 


7*2 


j. 0*05 


0-05310 


0-12265 


0-19488 


0-26748 


0-34018 


0-41293 


0-77681 


o-io 


0-07903 


0-14170 


0-20493 


0-26832 


0-33176 


0-39524 


0-71277 


0'15 


009990 


0-15658 


0-21336 


0-27022 


0-32708 


0-38395 


0-66840 


0-20 


011631 


0-16781 


0-21931 


0-27083 


0-32234 


0-37386 


0-63145 


0-25 


0-12894 


0-17582 


0-22268 


0-26955 


0-31643 


0-36330 


0-59767 


0-30 


013835 


0-18096 


0-23361 


0-26627 


0-30895 


0-35163 


0-56510 


0-35 


0-14494 


0-18355 


0-22224 


0-26098 


0-29976 


0-33855 


0-53271 


0-40 


0-14905 


0-18384 


021878 


0-25380 


0-28888 


0-32399 


0-49995 


0-45 


0-15097 


0-18212 


0-21344 


0-244S8 


0-27641 


0*30800 


46652 


0*50 


0-15099 


0-17860 


0-20642 


0-23439 


0-26247 


0*29065 


0-43232 



/ = 22° 30'. 





6=0 


c=0-l 


c=0-2 


c=0-3 


c=0-4 


o=0-5 


c=10 


a 


P 


P 


P 


P 


P 


P 


P 




r2 


r°- 


r> 


r 


r* 


r- 


ri 


0-05 


0-06102 


0-13346 


0-20621 


0-27899 


0-35178 


0-42458 


0-78857 


o-io 


0-08700 


0-15053 


0-21407 


0-27760 


0-34113 


0-4O466 


0-72233 


0-15 


0-10877 


0-16567 


0-22257 


0-27947 


0-33638 


0-39328 


067778 


0-20 


0-12635 


0-17785 


022936 


0-28087 


0-33239 


0-38391 


0-64150 


0-25 


0-14037 


0-L8716 


0-23399 


0-28082 


032767 


037453 


0-60886 


0-30 


0-15129 


0-19381 


0-23640 


0-27902 


0-32166 


0-36432 


0-o7773 


0-35 


0-15948 


0-19804 


0-23669 


0-27540 


0-31415 


0-35292 


0-54700 


0-40 


0-16525 


0-20005 


0*23497 


0-26999 


0-30506 


0-34017 


0-51608 


045 


0-16883 


0-20005 


0-23141 


0-26289 


0-29444 


0-32604 


048460 


0-50 


017047 


0-19824 


0-22617 


0-25423 


0-28238 


0-31060 


0-45241 



i = 30°. 





c=0 


c=0-l 


c=0-2 


c=0-3 


c=0-4 


c=0-5 


o 10 


a 


P 


P 


P 


P 


P 


P 


P 




r 2 


r2 


r 


T^ 


r> 


r 


r-2 

0-81731 


0-05 


0-09355 


0-16408 


0-23605 


0-30S45 


0-38101 


045365 


o-io 


0-11297 


0-17592 


0-2X922 


0-30263 


0-36609 


042957 


n-74711 t 


0-15 


0-13295 


0-18962 


0*24640 


0-30823 


0-36009 


0-41696 


0-701X8 1 


020 


0-15038 


0-20172 


0-25314 


0-30459 


0-3oH06 


0-40755 


0-66506 


0-25 


0-16493 


0-21160 


0-25834 


0-30513 


035193 


39876 


0-6H299 


030 


0-17673 


021917 


0-26170 


0-80427 


0-3468S 


038^51 


0-60282 


0-?5 


0-18599 


0-22452 


0-26314 


0-30182 


034055 


037930 


0-57332 


040 


0-19293 


0-22777 


0-26271 


02977 3 


0-33280 


0-367W1 


054380 | 


045 


0-19774 


0-22906 


0-26050 


0-29202 


0-32361 


0-35524 


0-51885 


0-50 


0-20060 


0-22854 


0-25661 


0-28476 


0-31299 


0-34128 


0-48327 



HORIZONTAL THRUST OF AX ARCn. 

,=370 30'. 



693 





c=Q 


c=0l 


c=0-2 


c=0-3 


c=04 


c=0-5 


c=l-0 


a 


P 


P 


P 


P 


P 


P 


P 




re 


ri 


ri 


r J - 


r J - 


r2 


1* 


0-05 


0-14749 


0-21733 


0-28854 


0-36038 


0-43255 50490 


086784 


010 


0-15949 


0-22174 


0-28457 


034768 


0-41093 0-47426 


41 


0-15 


0-17605 


0-23233 


0-28886 


0-34553 


0-40226 


0-45904 


9-74322 


0-20 


0-19209 


0-24321 


029448 


034583 


0-39722 


0-44865 


0*70598 


025 


0-20627 


0*25282 


0-29948 


0-34619 


0-39294 


043972 


7882 


0-30 


0-21827 


0-26066 


0-30314 


0-34568 


0-38825 


0-43085 


0*64406 


035 


22805 


0-26659 


0-30521 


0-34388 


0-38259 


42133 


0*61529 


0*40 


0-23570 


0-27060 


0-30558 


0-34062 


0-37571 


041083 


0-58673 


0-45 


0-24130 


0-27275 


0-30427 


0-33586 


036749 


039916 


0-55787 


050 


0-24499 


0-27312 


0-30132 


0-32958 


0-35789 


0-38625 


0*52845 



=45°. 





c=0 


c=01 


c=0.2 


c=0-8 


c=0-4 


c=0-5 


c=l-0 


a 


P 


P 


P 


P 


P 


P 


P 




r> 


re 


ri 


rl 


r-' 


r2 


ri 


005 


0-23105 


0-30081 


037162 


0-44305 


0-51485 


0-58688 


0-94881 


010 


0-23318 


0-29507 


035754 


042034 


0-48333 


0*54646 




0-15 


0-24478 


0-30079 


0-35708 


041355 


0-47013 




0-81- 


0*20 


0-25819 


0-30915 0-36028 


0-41151 


0*46281 


0*51415 


0*77124 


0-25 


0-27104 


0-31752 


0-36410 


0-41H74 


0-45744 


50417 




0-30 


0-28248 


0-32486 


0-36731 


0-40981 


045286 


0-49193 




0-35 


0-29216 


0-33073 


0-36935 


0-40803 


044674 


18547 




040 


0-29997 


0-33494 


0-36998 


0-40506 


0-44O1C, 






0-45 


0-30589 


0-33745 


0-36907 


40072 


0*48240 


0*46412 


0*62294 


0-60 
1 


0-3091)6 


0-33824 


0-36657 


0-39494 




0*45177 


- 1 1 :• 







APPENDIX. 



TABLE IV. 
Mechanical Properties of the Materials of Construction. 



Xet'.—Tbe capitals affixed to the cumbers to this table refer to the feOoving authorities: 



B. Biriow. Report to tk$ 

tke Xarv. dec 
Be. Bcrao. 

Br. Be.idor. ArcJL Rydr. 
h:z. Bm aL 

I). W. Daniel! and Wheatetone, Report on «<• 



SoneJorVuBomfofl 
Fairbaim. 
B idgttns -. 



Kamaa 



jb arfl 



•odation ofScimoa, <**. 



v v . ► -; : £./.; • /..;;•-.-! 

v 

Ml ntonbet 

Pa load Pad r. 

B. Bondelet, ZJr< dV Air 

. 7:. an : 

Tr. TredsoJd. Aaoy on the Strmg* tf 

Cast Iron. 
W. Watson. 



Acacia Ene. erowth) 
Air (atmospheric). . 

Alder . 

Antimony easl 
Anglo h w 



Bean Toaqum) 

Berch 

Birch (common) 

B ne I m a . 






:- 



- 

Do. pale red . 
Brick-work 




- 

DO (C r 

Do. (%Hretoa) . 
rJatterty) . 
• 
Do. (Weah stone) 
Do. (coke). . 
Do. (Wefch slaty) 
Da (Darby, Banal) 

Do. c 






: > 
-. n 

MS B 

I SI M 

: .- y. 
48 BL 
MM 

. -.' :. 
l m 3. 



•0 I C. 

-. i a i 

l-?19Br. 

1-000 Ml 

: . 4 Mi 
: : M: 
: OH Ml 
101 Mt 

! I I Ml 
: ftfl Ml 



"-:■- 

:: - 

4- " 

4- : :. 
-■■'. 



:::: •• b 






4-: -- 



- 
: - !■ 



4 : 



-.74 • 



n • 



-- -- 

• -- 

i ii 

I IB 



'«:; 






909 H 
(OMi BL 

rrssH. i 

M0I hytt | 

CSH B 

11663 H. 



::. I I 



IfJH ■ 



■■:-• }■: 



10E01 BL 
mOBSe. 



MM Ba 
IMO hi 



mm R. 



- - :. 



5674 B 



MOM r. 



PROPERTIES OF MATERIALS OF CONSTRUCTION. 



G05 









Tenacity 


Crushing 




| 




Specific 


We ght of 


per 


force per 


Modulus of 


Modulus of 


Name? >( Materials. 


gravity. 


cubic foot 
in lbs. 


square inch 
in lbs. 


fquare inch 
in lbs. 


elasticity li. 


rupture S. 


Coal (Bonlavooneen) . 


1-436 Mt. 


89-75 










Do. (coke) . 


1-596 Mt. 


99 75 










Do. (Corgee) 


1-403 Mt. 


87-63 










Do. (coke) . 


1-656 Mt. 


103-50 










Do. (Staffordshire) . 


1-240 


78-12 










Do. (Swansea) . 


1-357 K. 


84-81 










Do. (Wigan) 


1-263 K. 


79-25 










Do. (Glasgow) . 


1-29D 


80-62 










Do. (Newcastle) 


1-257 K. 


78-56 










Do. (common cannel) 


1-232 K. 


77-00 










Do. (slaty cannel) 


1-426 K. 


89-12 










Copper (cast) . 


8-607 


537-93 


19072 








Do. (sheet) 


8-785 


549-06 










Do. (wiredrawn) 


8-878 


560-00 


61223 








Do. (in bolts) . 


. . 




48000 








Crab-tree . 


0-765 


'47-80 


• 


6499 H. 






Deal (Christiana mid- 














dle) .... 


0-698 B. 


4362 


12400 




1672000 B. 


9364 B. 


Do. (Memel middle) . 


0-590 B. 


36 87 






1535200 B. 


108S6 B. 


Do. (Norway spruce) 


0-340 


21-25 


' 17600 








Do. (English) . 


0-470 


29-37 


7000 








Earth (rammed) 


1-584 Pa. 


9900 










Elder .... 


0-695 M 


4343 


10230 


8467 H. 






Elm (seasoned) . 


0-5SS C. 


36 75 


18489 M 


10331 H. 


699340 B. 


6078 B. 


Fir (New England) . 


0-553 B. 


34-56 






2191200 B. 


6612 B. 




( 


' 11549 B. 


5748 H. 


1328S00 B. 


6648 B. 


Do. (Riga) . 


0-753 B. 


47-06 -j 


to 12357 B. 


to65S6H. 


869600 B. 


7572 B. 


Do. (Mar Forest) 


0-693 B. 


43-31 










Flint .... 


2-630 T. 


164-37 










Glass (plate) 


2 453 


15331 


9420 








Gravel 


1920 


12000 










Granite (Aberdeen) . 


2526 


164-00 










Do. (Cornish) . 


2*662 


166 30 










Do. (red Egyptian) . 


2654 


16580 










Hawthorn . 


0-91 Be. 


33-12 


10500 Be. 








Hazel .... 


0-86 Be. 


53 75 


18000 Be. 








Holly .... 
Horn of an ox . 


0-76 Be. 


47-5 


16000 Be. 








1-6S9 M. 


10556 


8949 








Hornbeam (dry) 


0-760 R. 


47-50 


20240 Be. 


7289 H. 






Iron (wrought Eng.) . 


7-700 


481-20 


25itons, La. 








Do. (in bars) 


( -7600 
\ to7-800 


475-50 
437-00 


25itons, La. 








Do. (hammered) 






3') tons,Bru. 








Do (Russian) in bars 


. 




27 tons, La. 








Do (Swedish) in bars 






32 tons, R. 








Do. (English) in wire 


}■ '■ 




\ 36 to 43 








l-10th inch diameter 




■j tons, Te. 








Do (Russian) in wire 


1 




( 60 to 91 
\ tons, La. 








l-2f)th to l-30th inch 


r 










diameter . 


j 










Do. rolled in sheets 














and cut lengthwise. 






14 tons, Mi. 








Do. cut crosswise 






18 tons, MI 








Do. in chains, oval 














link<, 6 inches clear, 














iron y 2 inch diam. . 


. 


. 


21$ tons, Br 








Do. (Brunton's) with 

stay across link 
I ion." cast (old Park) . 
Do. (Adelphi) . 
Do (Alfreton) . 
Do. (scrap) 




. 


25 tons, B. 




17686400 T 


1 
• 
■i t(l 1' 


Do. (Carron, No. 2 

cold blast) 
Do. (hot blast) . 


7-oor, n. 

7-04(i n. 


441-62 
440-87 






10086000 H 


B85M 11 • 



•The numbers marked thus ' are calculated from the 0*P< rim< I 



c>oc> 



APPENDIX. 









Tenacity 


Cruihinj 






8prciSe 


Wri;hl nt I 


prr 




Mrtrrtatf 


Moati.n. of 


Ninwi of M.trrUU 


gravity. 


cubic foot 
in lb*. 


tuoair inch 

ii. lb. 


•qu«rr inrb 
in lb. 


»1». licit) E. 


rapture S. 


Iron, c 














Carroll cold blast . 


7094 F. 


448-37 


1420'! H. 


115442 H. 


16246966 F. 85980 F.* 


blast 


7 056 P. 


44100 


17755 11. 






Do. Devon. No. 3, 












cold blast) 


7-296 II. 


455 93 


. 




2290770ft 11. 8 


Do. (hot blast) . 


7429 II. 


451-61 


291o7 H 


145435 II. 


22473050 II. 4848TH.' 


Do (Buffery, No. 1, 












cold blast ) 


7079 II 


442-43 


1740.; II. 


93366 H. 


15351300 II B75 8 If * 


■ blast) . 


6 998 U. 


43737 


13434 II. 


86897 11. 


II. 35816 II* 


Do. (Coed Talon, No. 












2, cold blast 


6-955 F. 


484-06 


1S555II. 


817T0 II. 


14313500 F 881 4 F * 


Do. (hot blast) . 


6-903 F. 


435-50 


10676 H. 




14822609 F. 


33145 1 .* 


Do. (Coed Talon, No. 












| 


8, cold blast) . 


7-194 F. 


449-62 


, 


. 


17102000 F. 


43541 F* 


Do. (hot blast ) . 


6-970 F. 


435-62 


, , 




14707900 F. 


41.159 F.* 


Do. (KL-icar, No. 1, 














cold blast) 


7-080 F. 


439-37 


t 


e 


139S1000 F. 




Do (Milton, No. 1. 














hot blast). 


6-976 F. 


436-00 


, 


. 


11974500 F. 


25552 F .* 


Do. (Muirkirk, No. 1, 














cold blast) 


7-113 F. 


44456 


. . 




14008550 F. 


85928 F.*! 


Do. (hot blast) . 


6-953 F. 


434 56 




. 


18294400 F. 


83S50F.* 


Ivory .... 


1-826 P. 


11412 


16-626 








Laburnum . 


092 Be. 


57 50 


10500 Be. 








Lance-wood 


1-022 


6387 


24696 








Larch .... 


0522 B. 


3262 


10220 B.-j 


8201 H. ) 
(green) f 


697600 B. 


4992 B. 


Do. (second specimen) 


0-560 B. 


85-00 


6900 B. -j 


(dry) i 


1052800 B. 


6S94B. 


Leid (cast English) . 


11 446 M. 


717 45 


1824 Be. 


720000 Tr. 




Do (milled sheet) . 


11407 T. 


712-93 


83-25 Tr. 








Do. (wire) . 


11-817 T. 


70618 


2581 M. 








Lignum vitae [ous) 


1-280 


76-25 


11800 M. 








Limestone (arenace- 


■i -742 


17137 










Do. (foliated) . 


2-S37 


17731 










Do. (white fluor) . 


8156 


197-25 










Do. (green) . 


8182 


198-87 










Lime-tree . 


0-760 


47-50 


23500 Be. 








Lime (quick) 


0-453 Br. 


52-65 










Mahogany (Spanish) . 


o-soo 


50-00 


16500 


6198 n. 






Maple (Norway) 


0-798 


49-56 


10554 








Marble (white Italian) 


2-635 H. 


18487 


. . 


, 


2520000 T. 


1062 


Do. (black Galwav) . 


2495 II . 


10$ -25 






. 


2664 


M.rcurv (at 82 


13-619 












Do. (at 60°) . 


13-580 


-4-70 










Marl . . J 


1-600 


100-00 










to 2-877 T. 


11S-31 










Mortar 


1 751 Br. 


107-13 


50 

I 


4684 II. 


/ 




Oak (English) . 


0-984 B. 


55-37 


17-300 mJ 


95 9 II. 

(dry) 
4231 11 


V 1451200 B. 
1 


10082 B. 


Do. (Canadian). 


0-872 B 


54-50 


10-253 M.-^ 


9509 H. 
(.dry) 


-2145500 B. 


10596 B. 


Do. (Dantzic) . 


0756 B. 


47-24 


12-760 


1191200 B. 


6742 B. 


Do. (Adriatic) . 


0-993 B. 


62-06 




, 


974400 B. 


6298 B. 


Do. (African middle). 


0-972 B. 


60 75 


• 


• 


2288200 B. 


18566 B. 


Pear-tree . 


0461 M. 


41 -.11 




751S H. 




1 


Pine (pitch. 


0-660 B. 


4125 


*751SM! 




12266 


9792 B. 


Da (red) . 


0-657 B. 


4106 




5375 H. 


1840000 B. 


6946 B. 


Do. (Amer. yellow) . 


0-461 C. 


28 81 




5445 II. 


1600' 




Plane-tree . 


0-64 Be. 


4000 


11700 Be. 
( 


9367 H. 






Plum-tree . 


0-786 M. 


49 06 


11-351^ 


8657 H. 

(wet) 

8107 II. 












I 






Poplar. 


0-3SS M. 


23-93 


■Be.-^ 


5124 H. 

(dry) 






Pozzolano . 


2 677 K. 


169-37 


f 






1 Sand (river) 


1-SS6 


117-87 










-.tine (CTeen , l . 


2-574 K. 


1 163-S7 











PROPERTIES OF MATERIALS OF CONSTRUCTION. 



697 







Tenacity 


Crushing 




Names of mater ab. g S ^ c 


Weight of 
1 cubic foot 


Pef - V 
squari inch 

in lbs. 


fore- ), r r 

jqiiar^ inch 

■a lb,. 


Modnloiof 'jJodulutof 
elasticity E. j rapture S. 


Shinsjle - - - 1 424 Pa. 


89-00 








Silver (standard) - 10 312 T. 


644-50 


40902 M 






Slate (Welsh) - - 2S83 


180-50 


12800 


- 


15300000 Tr. 11700 Ee. 


do. (Westmoreland) - 2-791 W. 


174-48 


- 


- 


12900000 Tr. 




do. (Valcntia) - - 2SS0 Ee. 


180-00 








5226 Ee. 


do. (Scotch) - ... 




9600 


- 


15700000 Tr. 




Steel (soft. - - 7-780 


486-25 


120000 








do. (razor-tempered) - 7S40 


49000 


150000 




29000000 Y. 




Stone ■ Ancaster) - 21S2D.W. 


136-37 










do. Barnack - - 2-C90D.W. 


130 62 










do. Binnie - - 2194 D.W. 


137-12 










do. Bolsover - - 2-316 D.W. 


144-75 










do. Box - - 1-689 D.W 


114-93 










do. Bramham Moor - 2'008 D.W. 


12550 










do. Brodsworth - 2093 D.W. 


130-81 










do. Cadeby - - 1-951 D.W. 


121-93 










do. Caithness - - 2-764 Ee. 


172-75 


- 


- 


- 


5142 Ee. 


do. Craigleith- - 2266 D.W. 


141-62 










do. Chilmark (A) - 2 366 D.W. 


147-87 










do. Chilmark (B) - 2 333 D.W. 


148 93 










do. Chilmark (C) - 2-481 D.W. 


155-06 










do. Darby Dale (Stan- 












cliffe) - - 2-623 D.W. 


164-25 










do. G-iffneuk - - 2 230 D.W. 


139-37 










do. Gunbarrel (Stanley) 2 260 D.W. 


141-25 










do. Ham Hill- - 2 260 D.W. 


14125 










do. Havdon - - 2 "040 D.W. 


127-50 










do. He'ddon - - 2 229 D.W. 


139-31 










do. Hildenly - - 2-098 D.W. 


13112 










do. Hookstone - 2253 D.W. 


140 81 










do. Hnddlestone - 2 147 D.W. 


134-18 










do. Little Hulton - 2357 H. 




- 


• 


• 


774 n. 


do. Kenton - - 2247 D.W. 


140-43 










do. Ketton - - 2-045 D.W. 


127-81 










do. Ketton rag - 2 490 D.W. 


15562 










do. Mansfield, or Lind- 












ley'sred - - 2-833 D.W. 


14612 










do. white - - 2-277 D.W. 


142 81 










do. MorlevMoor - 2-058 D.W. 


123-31 










do. Park Nook - '213S D.W. 


133-62 










do. Park Bpring - 2321 D.W. 


14506 










do Portland (Way croft: 












Quarrv - - j 2145 D.W. 


134-06 










do. Kedsate - - 12 239 D.W. 


139-93 










do. Eoach Abbey - |2134D.W. 


133-37 








2853 H. 


do. Rochdale - - 2-577 H. 


16106 








do. Stanley - 


2 227 D.W. 


18918 










do. Taynton - 


2-103 D.W. 


131-48 










do. Totterwhoe 


2-891 D.W. 


118-18 










do. Jackdaw crag 


2070 D.W. 


129 37 








1116 n. 


do. Yorkshire flag - 


2-320 H. 


14000 








2010 Be. 


do. Green Moor 


2534 Ee. 


158 87 


• 


" 


* 




Sycamore 


0-69 Be. 


43-12 


18000 Be. 








Teak (dry) 


0-657 C. 


41-06 


15000 B. 


12101 IL 


2414400 B. 


14772 B. 


Tile icommon) - 
Tin (cast) 


1-815 Br. 
7-291 Tr. 


113-48 
455 68 


5822 M. 


- 


460S000 Tr. 




Water (sea) 


1 027 T. 


64-18 










do. (rain) 
Walnut - 
Whalebone 
Willow (dry) 


1000 
0-671 M. 

0-890 


62 50 
4193 

24-37 


8130 M. 
T68T 

14000 Be. 


6645 II. 


820000 Tr. 




Yew .(Spanish) - 


0-807 M. 


5048 


8000 Be. 








Zinc 


7-023 W. 


489-96 


- 


- 


13630000 Tr. 





mined li.v tin- <-.Minn 

lion 



in the hist column of ihe above 

furm part of a more extended inquiry, which, 

J„^wn«i J3E»BSttBrX& SSBtt 

Jlnmtration.i of Mechanics, p. 4U2.) 



698 



APPENDIX. 



TABLE V. 



Useful Numbers. 



It . . 


=3-1415927 


Log. a . 


, =0-4971499 


Log.* 


=1-1447299 


1 

7t 

rt 2 . . 


. =0-3183099 
=9-8696044 


1 

*i- 2 " " 


=0-1013212 


1 

V2 • 


=1-7724538 
=05641896 
=1-4142136 



V2 ' 


. =0-7071068 


rti/2 • 


. =4-4428829 


Tt 
71 ' 


. =2-2214415 


V% 


. =0-4501582 


7i 




y/l 


. =1-2533141 


VI- 


. =0-7978846 



a =2-7182818 

Log. s =0-4342945 

Modulus of common logarithms =-434294482 

Log. of ditto =9-6377843 

g =32-19084 

|/<7 =567363 

Log. g =1-5077222 

Inches in a French metre =39*37079 

Log. of ditto =1-5951741 

Feet in ditto =3-2808992 

Log. of ditto =0-5159929 

Square feet in the square metre =10*764297 

Acres in the Are =0*024711 

Lbs. in a kilogramme =2-20548 

Log. of ditto =0-3435031 

Imperial gallons in a litre =0*2200967 

Lbs. per square inch in 1 kilogramme per square millimetre =1422 

Cwts. ditto, ditto =12*7 

Volume of a sphere whose diameter is 1 . . . =0*5235988 

Arc of 1° to rati. 1 =0*017453293 

Arc of 1' to rad. 1 =0*000290888 

Arc of 1" to rad. 1 =0*000004848 

Degree in an arc whose length is 1 =57*295780° 

Grains in 1 oz. avoirdupois =437i 



Grains in 1 lb. ditto .... 






=7000 


Grains in a cubic inch of distilled water, Bar. 3 


in., Th. 62° =252-458 


Cubic inches in an ounce of water . 


. =1-73298 


Cubic inches in the imperial gallon . 






=277-276 


Feet in a geographical mile 






= 6075-6 


Log. of ditto 






=3-7835892 


Feet in a statute mile .... 






=5280 


Log. of ditto 






=3-7226339 


Length of seconds' pendulum in inches . 






=39-19084 


Cubic inches in 1 cwt. of cast iron . 






=430-25 


— Bar iron . 






=397-60 


— Cast brass 






=368-88 


— Cast copper 






=352-41 


— Cast lead . 






=272-80 


Cubic feet in 1 ton of paving stone . 






=14-835 


— Granite . 






=13-505 


— Marble . 






=13-070 


— Chalk . 






=12-874 


— Limestone 






=11-273 


— Elm 






= 64-460 


— Honduras mahogany 






=64-000 


— Mar Forest fir 






=51-650 


— Beech . 






=51-494 


— Kiga fir . 






=47-762 


— Ash and Dantzic oak 






=47-158 


— Spanish mahogany 






=42-066 


— English oak . 






=36205 


To find the weight in lbs. of 1 foot of common rope, mult 


i- 


ply the square of its circumference in inches by 


•044 to -04G 


Ditto for a cable 






•027 



Note. — The numerical values of the function of it in this table were calcu- 
lated by Mr. Goodwin. These, together with the numbers of cubic inches ami 
feet per cwt. or ton of different materials, are taken from the late Dr. Gregory's 
excellent treatise, entitled Mechanics for Practical Men. The other numbers 
of the table are principally taken from Mr. Bnbbage's Tables of Logarithms 
and the Aide Memoire of M. Morin. 



THE END. 



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